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

We have, $\displaystyle {(1+2x+3{x}^{2})}^{6}=\sum _{r=0}^{6}{ }^{6}{C}_{r}{(2x+3{x}^{2})}^{r}$(1+2x+3x2)6=r=0∑6​6Cr​(2x+3x2)r

$${=}^{6}{C}_{0}{+}^{6}{C}_{1}(2x+3{x}^{2}){+}^{6}{C}_{2}{(2x+3{x}^{2})}^{2}+\ldots {+}^{6}{C}_{6}{(2x+3{x}^{2})}^{6}$$=6C0​+6C1​(2x+3x2)+6C2​(2x+3x2)2+…+6C6​(2x+3x2)6

$\therefore$∴ Coefficient of ${x}^{2}=18+60=78$x2=18+60=78

Again, coefficient of ${x}^{2}$x2 in ${(1-4{x}^{2})}^{6}=-24$(1−4x2)6=−24

Constant term in $({(1+2x+3{x}^{2})}^{6}+{(1-4{x}^{2})}^{6})$((1+2x+3x2)6+(1−4x2)6) is $1+1=2$1+1=2.

Thus, the required coefficient of ${x}^{2}=2$x2=2 (coefficient of ${x}^{2}$x2) in $({(1+2x+3{x}^{2})}^{6}+{(1-4{x}^{2})}^{6})-$((1+2x+3x2)6+(1−4x2)6)−constant term in $({(1+2x+3{x}^{2})}^{6}+{(1-4{x}^{2})}^{6})$((1+2x+3x2)6+(1−4x2)6)

$$=2(78-24)-2$$=2(78−24)−2

$=106$=106.