1. **State the problem:** We need to evaluate the integral $$\int x^2 e^x \, dx$$. 2. **Formula and method:** We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = x^2 \quad \Rightarrow \quad du = 2x \, dx$$ $$dv = e^x \, dx \quad \Rightarrow \quad v = e^x$$ 3. **Apply integration by parts:** $$\int x^2 e^x \, dx = x^2 e^x - \int 2x e^x \, dx$$ 4. **Evaluate the remaining integral:** Again use integration by parts on $$\int 2x e^x \, dx$$. Let: $$u = 2x \quad \Rightarrow \quad du = 2 \, dx$$ $$dv = e^x \, dx \quad \Rightarrow \quad v = e^x$$ 5. **Apply integration by parts again:** $$\int 2x e^x \, dx = 2x e^x - \int 2 e^x \, dx = 2x e^x - 2 e^x + C$$ 6. **Substitute back:** $$\int x^2 e^x \, dx = x^2 e^x - (2x e^x - 2 e^x) + C = x^2 e^x - 2x e^x + 2 e^x + C$$ 7. **Factor the expression:** $$= e^x (x^2 - 2x + 2) + C$$ **Final answer:** $$\int x^2 e^x \, dx = e^x (x^2 - 2x + 2) + C$$