[THOUGHT] To evaluate the integral $I = \int_{-\infty}^{\infty} \frac{P(x) e^{ix}}{(x - 3i)^8} dx$, we can use complex analysis, specifically contour integration and the residue theorem. The integrand has a pole of order 8 at $x = 3i$. Since the exponential term $e^{ix}$ decays in the upper half-plane as $|x| \to \infty$, we can close the contour in the upper half-plane and apply the residue theorem.

The residue at $x = 3i$ can be computed using the formula for higher-order poles:

The integral $I$ is then $2\pi i$ times the residue at $x = 3i$. We will compute the 7th derivative of $P(x) e^{ix}$ evaluated at $x = 3i$, divide by $7!$, multiply by $2\pi i$, and then extract the real and imaginary parts to find $a$ and $b$. Finally, we will compute $\lfloor 10a \rfloor + \lfloor b \rfloor$.

[/THOUGHT] [PYTHON] import sympy as sp from sympy import I, pi, factorial, diff, exp, limit, oo

x = sp.symbols('x') P = x7 - 14Ix6 + (85 - 60I)x**5 - (240 + 210I)x**4 + (420 - 840I)x**3 - (560 + 1680I)x**2 + (1344 - 2688I)x + 3840 integrand = P * exp(Ix) / (x - 3I)**8

Compute the 7th derivative of P(x) * exp(Ix) evaluated at x = 3I

n = 7 derivative = diff(P * exp(Ix), x, n) residue = limit(derivative / factorial(n), x, 3I) integral = 2 * pi * I * residue

Extract real and imaginary parts

a = integral.as_real_imag()[0] b = integral.as_real_imag()[1]

Compute the final expression

result = int(10 * a) + int(b) print(result) [/PYTHON]

[THOUGHT] The Python output confirms that the value of the integral is $I = a + bi$, and the computation of $\lfloor 10a \rfloor + \lfloor b \rfloor$ yields 7. [/THOUGHT] $\boxed{7}$