1. **State the problem:** Differentiate the function $$y = (x^2 + 4x + 6)^5$$ using the chain rule. 2. **Recall the chain rule formula:** If $$y = [u(x)]^n$$, then $$\frac{dy}{dx} = n[u(x)]^{n-1} \cdot \frac{du}{dx}$$. 3. **Identify the inner function and outer function:** - Inner function: $$u = x^2 + 4x + 6$$ - Outer function: $$y = u^5$$ 4. **Differentiate the inner function:** $$\frac{du}{dx} = 2x + 4$$ 5. **Apply the chain rule:** $$\frac{dy}{dx} = 5(u)^{4} \cdot (2x + 4)$$ 6. **Substitute back the inner function:** $$\frac{dy}{dx} = 5(x^2 + 4x + 6)^4 (2x + 4)$$ 7. **Final answer:** $$\boxed{\frac{dy}{dx} = 5(x^2 + 4x + 6)^4 (2x + 4)}$$