$[9]10.($a$)$ Derive the Maclaurin series of $f(x)=x^2{e}^{2x}$

$($HINT: start with the series for $e^z$ where $z=2x).$

$($b$)$ Use differentiability of power series to find the sum $F(x)={\displaystyle \underset{1}{\overset{}{}}}\frac{x^{2n}}{n}$ within its radius of convergence.