1. **State the problem:** Find the derivative of the function $f(x) = x(a x^2 + b x + 3)$. 2. **Rewrite the function:** Distribute $x$ to get $f(x) = a x^3 + b x^2 + 3 x$. 3. **Recall the derivative rules:** The derivative of $x^n$ is $n x^{n-1}$. The derivative of a sum is the sum of derivatives. 4. **Differentiate each term:** $$\frac{d}{dx}(a x^3) = 3 a x^2$$ $$\frac{d}{dx}(b x^2) = 2 b x$$ $$\frac{d}{dx}(3 x) = 3$$ 5. **Combine the results:** $$f'(x) = 3 a x^2 + 2 b x + 3$$ **Final answer:** $$\boxed{f'(x) = 3 a x^2 + 2 b x + 3}$$