Using the Maclaurin series, $e^x={\displaystyle \underset{k=0}{\overset{}{}}}\frac{x^k}{k!},$ compute the third order Taylor polynomial centered centered at $a=0$ for the function

$f(x)=x^2{e}^x$

$($a$)\frac{x^2}{2}+\frac{x^3}{6}$

$($b$)1+x+\frac{x^2}{2}+\frac{x^3}{6}$

$($c$)x^2+x^3$

$($d$)\frac{x^3}{6}$

$($e$)x^3$