Answer
\[-e^{-x}[x^2+2x+2]+C\]
Where $C$ is constant of integration
Work Step by Step
Let \[I=\int x^2e^{-x}dx\]
Using integration by parts
\[I=x^2\int e^{-x}dx-\int\left[(x^2)'\int e^{-x}dx\right]dx\]
\[I=-x^2e^{-x}+2\int xe^{-x}dx\]
Using integration by parts
\[I=-x^2e^{-x}+2\left[x\int e^{-x}dx-\int \left((x)'\int e^{-x}dx\right)dx\right]\]
\[I=-x^2e^{-x}+2\left[-x e^{-x}+\int e^{-x}dx\right]\]
\[I=-x^2e^{-x}-2xe^{-x}-2e^{-x}+C\]
Where $C$ is constant of integration
\[I=-e^{-x}[x^2+2x+2]+C\]
Hence ,
\[I=-e^{-x}[x^2+2x+2]+C\]
