[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: [ \text{Res}(f, 3i) = \frac{1}{7!} \lim_{x \to 3i} \frac{d^7}{dx^7} \left( (x - 3i)^8 \frac{P(x) e^{ix}}{(x - 3i)^8} \right) = \frac{1}{7!} \lim_{x \to 3i} \frac{d^7}{dx^7} \left( P(x) e^{ix} \right). ]

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}