1. **State the problem:** Factor the polynomial $5x^3 + 10x^2 - 20x - 40$. 2. **Identify common factors:** Notice that each term is divisible by 5. Use the distributive property to factor out 5: $$5x^3 + 10x^2 - 20x - 40 = 5(x^3 + 2x^2 - 4x - 8)$$ 3. **Factor the cubic inside the parentheses:** Group terms to factor by grouping: $$x^3 + 2x^2 - 4x - 8 = (x^3 + 2x^2) + (-4x - 8)$$ 4. **Factor each group:** $$x^2(x + 2) - 4(x + 2)$$ 5. **Factor out the common binomial:** $$\cancel{(x + 2)}(x^2 - 4)$$ 6. **Recognize difference of squares:** $$x^2 - 4 = (x - 2)(x + 2)$$ 7. **Write the full factorization:** $$5(x + 2)(x - 2)(x + 2) = 5(x + 2)^2(x - 2)$$ **Final answer:** $$\boxed{5(x + 2)^2(x - 2)}$$