Solution
= Solution
Apply the <Leibniz rule> for the $n$th <derivative> of a product. Only the first three terms survive, because all <derivatives> of $x^2$ of order greater than two vanish and every <derivative> of $e^x$ is $e^x$:
$$
\frac{d^n}{dx^n}(x^2e^x)=e^x\left[\binom n0x^2+\binom n1(2x)+\binom n2(2)\right]=e^x[x^2+2nx+n(n-1)].
$$
At $n=100$ this gives
$$
\boxed{\frac{d^{100}}{dx^{100}}(x^2e^x)=e^x(x^2+200x+9900).}
$$