1. **Problem:** Factor the polynomial $$y^6 + 10y^5 - 24y^4$$ completely. 2. **Formula and rules:** To factor polynomials, first look for the greatest common factor (GCF). Then factor the remaining polynomial if possible. 3. **Step 1:** Find the GCF of all terms. $$y^6, 10y^5, -24y^4$$ The smallest power of $y$ is $y^4$, so GCF is $$y^4$$. 4. **Step 2:** Factor out $$y^4$$: $$y^6 + 10y^5 - 24y^4 = y^4(y^2 + 10y - 24)$$ 5. **Step 3:** Factor the quadratic inside the parentheses. We look for two numbers that multiply to $$-24$$ and add to $$10$$. These numbers are $$12$$ and $$-2$$. 6. **Step 4:** Write the factorization: $$y^4(y + 12)(y - 2)$$ 7. **Final answer:** $$\boxed{y^4(y + 12)(y - 2)}$$