<s>[INST] Find coefficient of $x^2$ in a complicated expansion 
Find the coefficient of $x^2$ in the expansion of $(4-x^2)[(1+2x+3x^2)^6+(1+4x^3)^5]$

I noticed that since we only care about the coefficient of $x^2$, only the coefficients of $x^2$ inside the square brackets, as well as the constant terms, will matter. From there I am stuck when it comes to expanding $(1+2x+3x^2)^6$ as I can't see an easy way to use the binomial theorem (for example by factorising $1+2x+3x^2$, which I can't do as there are no real roots). Is there a way to find the coefficient of $x^2$ in this expansion (specifically $(3x^2+2x+6)^6$ as I can get the answer from there)?
 [/INST] For a question like this cheating is legal :)
$\left(4-x^2\right) \left[\left(4 x^3+1\right)^5+\left(3 x^2+2 x+1\right)^6\right]=-1024 x^{17}+4096 x^{15}-2009 x^{14}-2916 x^{13}+1718 x^{12}+1844 x^{11}+15417 x^{10}+31144 x^9+33976 x^8+29832 x^7+20681 x^6+10488 x^5+4302 x^4+1428 x^3+310 x^2+48 x+8$
The answer is $310$
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