1. **State the problem:** Find the derivative $P'(x)$ of the function $P(x) = x^3 e^x$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$ 3. **Identify functions:** Here, $f(x) = x^3$ and $g(x) = e^x$. 4. **Compute derivatives:** - $f'(x) = 3x^2$ - $g'(x) = e^x$ 5. **Apply product rule:** $$P'(x) = f'(x)g(x) + f(x)g'(x) = 3x^2 e^x + x^3 e^x$$ 6. **Factor common terms:** $$P'(x) = e^x (3x^2 + x^3) = e^x x^2 (3 + x)$$ 7. **Compare with options:** The correct derivative matches option (a): $$P'(x) = e^x (x^3 + 3x^2)$$ **Final answer:** $P'(x) = e^x (x^3 + 3x^2)$