$\int(x+1)^2 e^x d x$ is equal to

$x e^x+C$ $x^2 e^x+C$ $(x+1) e^x+C$ $\left(x^2+1\right) e^x+C$

$\left(x^2+1\right) e^x+C$

We have, $I=\int(x+1)^2 e^x d x=\int e^x\left(x^2+2 x\right) d x+\int e^x d x$ $\Rightarrow I=e^x x^2+e^x+C=e^x\left(x^2+1\right)+C$