1. **Problem statement:** Find the derivative of the function $y = (1 + 3x + 4x^2)^{-3}$. 2. **Formula and rules:** Use the Chain Rule for derivatives of composite functions. If $y = [u(x)]^n$, then $y' = n[u(x)]^{n-1} \cdot u'(x)$. 3. **Identify inner function:** Let $u = 1 + 3x + 4x^2$. 4. **Compute $u'$:** $$u' = \frac{d}{dx}(1 + 3x + 4x^2) = 0 + 3 + 8x = 3 + 8x$$ 5. **Apply Chain Rule:** $$y' = -3(1 + 3x + 4x^2)^{-4} \cdot (3 + 8x)$$ 6. **Final answer:** $$\boxed{y' = -3(1 + 3x + 4x^2)^{-4}(3 + 8x)}$$