[THOUGHT] To evaluate the integral $I = \int_{-\infty}^{\infty} \frac{P(x) e^{ix}}{(x - 3i)^8} \, dx$, we use the residue theorem from complex analysis. The integrand has a pole of order 8 at $x = 3i$. The integral is $2\pi i$ times the residue at this pole. The residue is given by $\frac{1}{7!}$ times the 7th derivative of $P(z) e^{iz}$ evaluated at $z = 3i$. We compute this derivative symbolically using sympy, evaluate it at $3i$, and then extract the real and imaginary parts to find $a$ and $b$. Finally, we compute $\lfloor 10a \rfloor + \lfloor b \rfloor$. [/THOUGHT] [PYTHON] import sympy as sp

z = sp.symbols('z', complex=True) I = sp.I pi = sp.pi

Define the polynomial P(z)

P = z7 - 14Iz6 + (85 - 60*I)z**5 - (240 + 210I)z**4 + (420 - 840I)z**3 - (560 + 1680I)z**2 + (1344 - 2688I)*z + 3840

Compute the 7th derivative of P(z) * exp(I*z)

f = P * sp.exp(I * z) seventh_derivative = sp.diff(f, z, 7)

Evaluate at z = 3I

seventh_derivative_at_3i = seventh_derivative.subs(z, 3*I)

Compute the integral I

integral = (2 * pi * I) * seventh_derivative_at_3i / sp.factorial(7)

Separate into real and imaginary parts and evaluate numerically

a = sp.re(integral).evalf() b = sp.im(integral).evalf()

Compute the result

result = sp.floor(10 * a) + sp.floor(b) print(result) [/PYTHON]

[THOUGHT] The tool output indicates the result $\lfloor 10a \rfloor + \lfloor b \rfloor = 7$. This is the final answer based on the computed integral value. [/THOUGHT] $\boxed{7}$