1. **Problem statement:** Find the derivative $f'(x)$ of the function $f(x) = x^3 + 2x^2 + 4x + 5$. 2. **Formula used:** The derivative of a function $f(x)$ is found by applying the power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ and the derivative of a constant is zero. 3. **Step-by-step differentiation:** - Derivative of $x^3$ is $3x^2$. - Derivative of $2x^2$ is $2 \times 2x = 4x$. - Derivative of $4x$ is $4$. - Derivative of constant $5$ is $0$. 4. **Combine all terms:** $$f'(x) = 3x^2 + 4x + 4$$ 5. **Explanation:** Each term is differentiated separately using the power rule, and constants vanish. The correct derivative matches option A. **Final answer:** $f'(x) = 3x^2 + 4x + 4$ (Option A).