1. **State the problem:** Factor the polynomial completely: $$-2p^3 + 12p^2 + 54p$$. 2. **Identify the greatest common factor (GCF):** Look at the coefficients and the variable powers. - Coefficients: -2, 12, 54. The GCF of 2, 12, and 54 is 2. - Variables: Each term has at least one $p$, so factor out $p$. 3. **Factor out the GCF:** $$-2p^3 + 12p^2 + 54p = -2p(p^2 - 6p - 27)$$ 4. **Factor the quadratic inside the parentheses:** We want to factor $$p^2 - 6p - 27$$. Find two numbers that multiply to $$-27$$ and add to $$-6$$. These numbers are $$-9$$ and $$3$$. 5. **Write the factorization:** $$p^2 - 6p - 27 = (p - 9)(p + 3)$$ 6. **Final factorization:** $$-2p^3 + 12p^2 + 54p = -2p(p - 9)(p + 3)$$ This is the complete factorization. **Answer:** $$-2p(p - 9)(p + 3)$$