1. **State the problem:** Factor the polynomial $$x^3 + 9x^2 + 14x - 24$$ completely given that $$x + 6$$ is a factor. 2. **Use synthetic division:** Since $$x + 6$$ is a factor, use $$-6$$ as the divisor in synthetic division. Set up synthetic division with coefficients: 1 (for $$x^3$$), 9 (for $$x^2$$), 14 (for $$x$$), and -24 (constant). 3. Perform synthetic division: - Bring down 1. - Multiply $$1 \times (-6) = -6$$, add to 9: $$9 + (-6) = 3$$. - Multiply $$3 \times (-6) = -18$$, add to 14: $$14 + (-18) = -4$$. - Multiply $$-4 \times (-6) = 24$$, add to -24: $$-24 + 24 = 0$$. The remainder is 0, confirming $$x + 6$$ is a factor. 4. The quotient polynomial is $$x^2 + 3x - 4$$. 5. **Factor the quadratic:** $$x^2 + 3x - 4$$ factors as $$(x + 4)(x - 1)$$ because $$4 \times (-1) = -4$$ and $$4 + (-1) = 3$$. 6. **Write the complete factorization:** $$x^3 + 9x^2 + 14x - 24 = (x + 6)(x + 4)(x - 1)$$ **Final answer:** $$(x + 6)(x + 4)(x - 1)$$