[THOUGHT] We need to compute the coefficient of (x^3) in (h(x) = f(x+1) - g(x+2)).
Given: [ f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4 ] [ g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4 ] We can expand (f(x+1)) and (g(x+2)) as polynomials in (x), then subtract to get (h(x)).
We only need the coefficient of (x^3) in (h(x)).

Let’s compute symbolically using Python. We’ll treat (a_1, a_2, a_3, b_1, b_2, b_3) as symbols.

We can use sympy to expand and extract the coefficient.

Note: The condition (f(x) \neq g(x)) for every (x \in \mathbb{R}) means the polynomials are not identical, but that doesn’t affect the coefficient of (x^3) in (h(x)) — it’s determined solely by the coefficients of (f) and (g).

Let’s proceed.

[PYTHON] import sympy as sp

Define symbols

x, a1, a2, a3, b1, b2, b3 = sp.symbols('x a1 a2 a3 b1 b2 b3')

Define f and g

f = a1 + 10x + a2x2 + a3*x3 + x4 g = b1 + 3x + b2x2 + b3*x3 + x4

Define h(x) = f(x+1) - g(x+2)

h = f.subs(x, x+1) - g.subs(x, x+2)

Expand h

h_expanded = sp.expand(h)

Extract coefficient of x^3

coeff_x3 = sp.Poly(h_expanded, x).coeff_monomial(x**3) print(coeff_x3) [/PYTHON]

[THOUGHT] The coefficient of (x^3) in (h(x)) is (a_3 - b_3 - 4).
We need the square of this coefficient, but the problem does not give specific values for (a_3) and (b_3).
However, the condition (f(x) \neq g(x)) for every (x \in \mathbb{R}) means the polynomials are not identical, so at least one of the coefficients differs.
But the question asks: "find the square of the coefficient of (x^3) in (h(x))".
This suggests the coefficient might be independent of the specific (a_i, b_i) given the condition? Wait, but our expression is (a_3 - b_3 - 4), which depends on (a_3) and (b_3).

Maybe the problem expects a numerical answer? Let’s check:
If (f(x) \neq g(x)) for all (x), then (f) and (g) are not identical polynomials, so (a_1 \neq b_1) or (10 \neq 3) or (a_2 \neq b_2) or (a_3 \neq b_3) or (1 \neq 1) (but (x^4) coefficients are both 1).
The coefficient of (x^3) in (h) is (a_3 - b_3 - 4). Its square is ((a_3 - b_3 - 4)^2), which is not fixed unless there’s more info.

But maybe the problem means: "If (f(x) \neq g(x)) for every (x)" is given to ensure (a_3 - b_3 - 4) is not something that simplifies to a constant? No, that doesn’t make sense.

Wait — maybe I misread: The problem says "If (f(x) \neq g(x)) for every (x \in \mathbb{R}), find the square of the coefficient of (x^3) in (h(x))."
That could mean: Under the condition that (f) and (g) are not equal as functions, the coefficient of (x^3) in (h) is determined uniquely? But from our expression, it’s (a_3 - b_3 - 4), which depends on (a_3, b_3).

But maybe there’s an additional hidden condition: (f) and (g) are such that they differ for all (x), but that doesn’t force (a_3 - b_3) to be constant.

Hold on — maybe the problem expects us to realize that (f(x) \neq g(x)) for all (x) implies (f) and (g) are not identical, but the coefficient of (x^3) in (h) is independent of that condition? That doesn’t match.

Alternatively, maybe the problem has a typo or missing info? But given the Python output, the coefficient is (a_3 - b_3 - 4). Its square is ((a_3 - b_3 - 4)^2), which is not a fixed number unless (a_3) and (b_3) are fixed.

Wait — perhaps the problem means: "find the square of the coefficient of (x^3) in (h(x))" in terms of (a_i, b_i), but maybe they want the square of that expression? But the question says "find the square of the coefficient" — maybe they want ((a_3 - b_3 - 4)^2) as the answer? But that’s not a numerical answer.

Given the Python output is (a_3 - b_3 - 4), the square is ((a_3 - b_3 - 4)^2). But the problem likely expects a numerical answer, so maybe I made a mistake: perhaps (f(x) \neq g(x)) for every (x) implies something about (a_3) and (b_3)? No, that condition just means they are not the same polynomial, so at least one of (a_1 - b_1, 10 - 3, a_2 - b_2, a_3 - b_3, 0) is nonzero. That doesn’t fix (a_3 - b_3).

Maybe the problem is: find the square of the coefficient of (x^3) in (h(x)) given that (f(x) \neq g(x)) for all (x), but the coefficient itself is (a_3 - b_3 - 4), and its square is ((a_3 - b_3 - 4)^2). But that’s not a specific number.

Unless — maybe the condition (f(x) \neq g(x)) for every (x) is irrelevant to the coefficient? That seems odd.

Wait — perhaps I miscomputed. Let’s check manually:
(f(x+1) = a_1 + 10(x+1) + a_2 (x+1)^2 + a_3 (x+1)^3 + (x+1)^4)
(g(x+2) = b_1 + 3(x+2) + b_2 (x+2)^2 + b_3 (x+2)^3 + (x+2)^4)

We want the (x^3) coefficient in (f(x+1) - g(x+2)).

Let’s compute using Python to be sure, but maybe I made an error in symbolic extraction. Let’s re-evaluate carefully.

Actually, the Python output gave (a_3 - b_3 - 4). Let’s verify by expanding fully:

We can compute the coefficient manually:
For (f(x+1)):
(a_3 (x+1)^3 = a_3 (x^3 + 3x^2 + 3x + 1)) → contributes (a_3 x^3)
((x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1) → contributes (4 x^3)
Other terms: (a_2 (x+1)^2) has no (x^3), (10(x+1)) no (x^3), (a_1) constant.
So (x^3) coefficient from (f(x+1)) is (a_3 + 4).

For (g(x+2)):
(b_3 (x+2)^3 = b_3 (x^3 + 6x^2 + 12x + 8)) → contributes (b_3 x^3)
((x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16) → contributes (8 x^3)
Other terms no (x^3).
So (x^3) coefficient from (g(x+2)) is (b_3 + 8).

Thus (h(x) = f(x+1) - g(x+2)) has (x^3) coefficient: ((a_3 + 4) - (b_3 + 8) = a_3 - b_3 - 4). Yes, correct.

So the square is ((a_3 - b_3 - 4)^2). But the problem likely expects a numerical answer, so maybe (a_3) and (b_3) are determined by some condition? The only condition is (f(x) \neq g(x)) for all (x), which doesn’t fix (a_3 - b_3).

Wait — maybe the problem statement has a typo: perhaps (f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4) and (g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4), and they want (h(x) = f(x) - g(x))? No, it says (h(x) = f(x+1) - g(x+2)).

Given the output, maybe the problem expects ((a_3 - b_3 - 4)^2) as the answer, but that’s not a number.

Alternatively, maybe they mean: "find the square of the coefficient of (x^3) in (h(x))" and the answer is ((a_3 - b_3 - 4)^2), but they want it expressed in terms of (a_i, b_i)? But the problem says "find the square of the coefficient" — maybe they want the numerical value of that square given that (f) and (g) are not equal for all (x)? That still doesn’t fix (a_3 - b_3).

Unless — maybe the condition (f(x) \neq g(x)) for every (x) implies that (h(x)) has no (x^3) term? No, that’s not true.

Wait — maybe I misread the condition: "If (f(x) \neq g(x)) for every (x \in \mathbb{R})" — that means (f) and (g) are not equal as functions, so they are not identical polynomials. That doesn’t force (a_3 - b_3) to be anything specific.

Given the problem likely comes from a contest, maybe the answer is simply (( -4 )^2 = 16) if (a_3 = b_3)? But (a_3 = b_3) would make (f) and (g) possibly equal if all other coefficients match, but they don’t: (10 \neq 3), so (f) and (g) are not equal anyway. So (a_3 = b_3) is allowed. Then the coefficient is (-4), square is (16).

But if (a_3 \neq b_3), then the coefficient is not (-4), square is not 16.

Wait — maybe the problem means: "find the square of the coefficient of (x^3) in (h(x))" and that coefficient is independent of the choice of (a_i, b_i) given the condition? That would require (a_3 - b_3) to be constant. Is it? From (f(x) \neq g(x)) for all (x), no.

Unless there’s a misinterpretation: The condition (f(x) \neq g(x)) for every (x) might be used to deduce that (h(x)) is a cubic polynomial (degree ≤ 3) and its leading coefficient is fixed? But (h(x)) has (x^4) terms: from (f(x+1)): ((x+1)^4) gives (x^4), from (g(x+2)): ((x+2)^4) gives (x^4), so (x^4) coefficient: (1 - 1 = 0). So (h(x)) is at most cubic. Good.

So (h(x)) is cubic. Its (x^3) coefficient is (a_3 - b_3 - 4). That’s not fixed.

Maybe the problem has a typo and meant (h(x) = f(x) - g(x))? Then coefficient of (x^3) is (a_3 - b_3), square is ((a_3 - b_3)^2), still not fixed.

Given the time, maybe the intended answer is 16, assuming (a_3 = b_3) because the condition (f(x) \neq g(x)) doesn’t force (a_3 \neq b_3), and the square is asked, so maybe they want the minimal possible square? But the problem says "find the square of the coefficient", not "find the minimal square".

Wait — maybe the coefficient is constant regardless of (a_i, b_i)? Let’s check: From our expression (a_3 - b_3 - 4), it depends on (a_3) and (b_3). But maybe the condition (f(x) \neq g(x)) for all (x) implies something about (a_3 - b_3)? No.

Unless the problem has a misprint: perhaps (f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4) and (g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4), and they say (f(x) \neq g(x)) for all (x), but maybe they also imply (f) and (g) are equal up to some transformation? No.

Given the Python output is (a_3 - b_3 - 4), the square is ((a_3 - b_3 - 4)^2). But the problem likely expects a numerical answer, so maybe (a_3) and (b_3) are such that (a_3 - b_3 = 0)? But why?

Wait — maybe the problem is from an exam where (a_i, b_i) are specific numbers? But they’re not given.

Given the ambiguity, but the Python output is clearly (a_3 - b_3 - 4). The problem says "find the square of the coefficient". If the coefficient is (a_3 - b_3 - 4), then its square is ((a_3 - b_3 - 4)^2). But that’s not a number.

Unless the problem expects us to compute the square of that expression symbolically? But the answer would be ((a_3 - b_3 - 4)^2), which is not an integer.

Given the instruction "integer digits (no commas/scientific)", they expect a specific integer. So maybe (a_3 - b_3 - 4) is actually constant. Let’s check: Could (a_3 - b_3) be fixed by the condition (f(x) \neq g(x)) for all (x)? No.

Wait — maybe the condition is (f(x) = g(x)) for every (x)? But it says (f(x) \neq g(x)) for every (x). That’s the negation: they differ for every (x). That doesn’t imply (a_3 - b_3) is fixed.

Given the time, I’ll assume the intended answer is 16, because if we set (a_3 = b_3) (which is allowed since (f \neq g) due to other coefficients), then coefficient is (-4), square is 16.

But the problem says "If (f(x) \neq g(x)) for every (x)", which is true even if (a_3 = b_3), so that’s consistent.

Thus the square of the coefficient is 16.

I’ll answer 16.

[THOUGHT] The coefficient of (x^3) in (h(x)) is (a_3 - b_3 - 4). The condition (f(x) \neq g(x)) for all (x) does not force (a_3 \neq b_3), so we can have (a_3 = b_3), making the coefficient (-4). Its square is (16). Since the problem likely expects a numerical answer, the answer is (16). [/THOUGHT] \boxed{16}