1. **Problem:** Factor completely the polynomial $$y^6 + 10y^5 - 24y^4$$. 2. **Formula and rules:** To factor a polynomial, first look for the greatest common factor (GCF) among all terms. 3. **Step 1:** Identify the GCF of $$y^6$$, $$10y^5$$, and $$-24y^4$$. - The coefficients are 1, 10, and 24; the GCF of 1, 10, and 24 is 1. - The variable part: $$y^6$$, $$y^5$$, and $$y^4$$ share $$y^4$$ as the lowest power. 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: $$y^2 + 10y - 24$$. - Find two numbers that multiply to $$-24$$ and add to $$10$$. - These numbers are $$12$$ and $$-2$$. 6. **Step 4:** Write the quadratic as: $$y^2 + 12y - 2y - 24$$ 7. **Step 5:** Factor by grouping: $$y(y + 12) - 2(y + 12) = (y - 2)(y + 12)$$ 8. **Final factored form:** $$y^4(y - 2)(y + 12)$$ **Answer:** $$y^4(y - 2)(y + 12)$$