The coefficient of $x^2$ in expansion of the product
(2-$x^2$).($(1 + 2x + 3x^2)^6$ + $(1-1 4x^2)^6$) is :

$\begin{array}{l}\left(2-x^{2}\right)\left[\left(1+2 x+3 x^{2}\right)^{6}+\left(1-4 x^{2}\right)^{6}\right] \\ =2\left[\left(1+2 x+3 x^{2}\right)^{6}+\left(1-4 x^{2}\right)^{6}\right]-x^{2}\left[\left(1+2 x+3 x^{2}\right)^{6}+\left(1-4 x^{2}\right)^{6}\right] \\ \Rightarrow \text { coefficient of } x^{2}=2 \cdot \text { coefficient of } x^{2} \text { in }\left[\left(1+2 x+3 x^{2}\right)^{6}+\left(1-4 x^{2}\right)^{6}\right] \\ - \text { constant term in }\left[\left(1+2 x+3 x^{2}\right)^{6}+\left(1-4 x^{2}\right)^{6}\right] \\ \left(1+2 x+3 x^{2}\right)^{6}=\sum_{r=0}^{6}{ }^{6} C_{r}\left(2 x+3 x^{2}\right)^{r} \\ ={ }^{6} C_{0}+{ }^{6} C_{1}\left(2 x+3 x^{2}\right)+{ }^{6} C_{2}\left(2 x+3 x^{2}\right)^{2}+\ldots \\ \Rightarrow \text { coefficient of } x^{2}=2(18+60-24)-2=106\end{array}$