<s>[INST] What is the coefficient of $x^2$ in the polynomial.. What is coefficient of $x^2$ in polynomial $(1-4x)^6(1+3x)^8$ ? 
I know how to do it for $(1-4x)^6$ for example, but how to do it for the product of two polynomials?
 [/INST] Let $f(x) = (1-4x)^6(1+3x)^8 = uv$, where $u = (1-4x)^6$ and $v=(1+3x)^8$ 
\begin{align*}
 f''(x) &= u'' v + 2 u'v' + v''\\
 &= 6 \cdot 5 \cdot (-4)^2 (1-4x)^4 (1+3x)^8 + 2 \cdot 6 \cdot (-4) \cdot 8 \cdot 3 (1-4x)^5 (1+3x)^7 \\
 & \qquad + 8 \cdot 7 \cdot (3)^2 (1-4x)^6 (1+3x)^6 
 \end{align*}
    Hence coefficient of $x^2$ is $\frac{1}{2!}f''(0) = (480 - 1152 + 504)/2 = -84 $
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