<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] Note that$(4-x^2)[(1+2x+3x^2)^6+(1+4x^3)^5]$ is equal to
$$4[(1+2x)^6+6\cdot 3x^2+1+o(x^2)]-x^2\cdot (1+1+o(x))\\
=4[(1+6\cdot 2x+\binom{6}{2}(2x)^2+18x^2+1+o(x^2)]-2x^2+o(x^2)\\
=8+48x+310x^2+o(x^2)$$
where we used the Binomial Theorem and the little-o notation.
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