1. **State the problem:** Find the derivative of the function $$y = (x^2 - 7)(x^2 + 4x + 2)$$. 2. **Formula used:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$ where $u = x^2 - 7$ and $v = x^2 + 4x + 2$. 3. **Find derivatives of each part:** $$u' = \frac{d}{dx}(x^2 - 7) = 2x$$ $$v' = \frac{d}{dx}(x^2 + 4x + 2) = 2x + 4$$ 4. **Apply the product rule:** $$y' = u'v + uv' = 2x(x^2 + 4x + 2) + (x^2 - 7)(2x + 4)$$ 5. **Expand each term:** $$2x(x^2 + 4x + 2) = 2x^3 + 8x^2 + 4x$$ $$ (x^2 - 7)(2x + 4) = x^2(2x + 4) - 7(2x + 4) = 2x^3 + 4x^2 - 14x - 28$$ 6. **Combine like terms:** $$y' = (2x^3 + 8x^2 + 4x) + (2x^3 + 4x^2 - 14x - 28)$$ $$= (2x^3 + 2x^3) + (8x^2 + 4x^2) + (4x - 14x) - 28$$ $$= 4x^3 + 12x^2 - 10x - 28$$ **Final answer:** $$\boxed{y' = 4x^3 + 12x^2 - 10x - 28}$$