1. **Stating the problem:** Find the derivative $y'$ of the function $y = (x^2 + 3)(x^4 - 5x)$. 2. **Formula used:** Use the product rule for derivatives: $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$ where $u = x^2 + 3$ and $v = x^4 - 5x$. 3. **Find derivatives of $u$ and $v$: $$u' = \frac{d}{dx}(x^2 + 3) = 2x$$ $$v' = \frac{d}{dx}(x^4 - 5x) = 4x^3 - 5$$ 4. **Apply the product rule:** $$y' = u'v + uv' = (2x)(x^4 - 5x) + (x^2 + 3)(4x^3 - 5)$$ 5. **Expand each term:** $$= 2x \cdot x^4 - 2x \cdot 5x + x^2 \cdot 4x^3 + x^2 \cdot (-5) + 3 \cdot 4x^3 + 3 \cdot (-5)$$ $$= 2x^5 - 10x^2 + 4x^5 - 5x^2 + 12x^3 - 15$$ 6. **Combine like terms:** $$= (2x^5 + 4x^5) + (-10x^2 - 5x^2) + 12x^3 - 15$$ $$= 6x^5 - 15x^2 + 12x^3 - 15$$ 7. **Rearrange terms in descending powers:** $$y' = 6x^5 + 12x^3 - 15x^2 - 15$$ **Final answer:** $$\boxed{y' = 6x^5 + 12x^3 - 15x^2 - 15}$$