1. **Problem statement:** Apply the distributive property to each expression to write an equivalent expression. 2. **Recall the distributive property:** For any expressions $a$, $b$, and $c$, we have $$a(b + c) = ab + ac$$ 3. **Apply to each expression:** **a.** $(x + 2)(x + 4)$ Use distributive property twice: $$= x(x + 4) + 2(x + 4)$$ $$= x^2 + 4x + 2x + 8$$ Combine like terms: $$= x^2 + 6x + 8$$ **b.** $(x + 2)(3x + 1)$ Distribute each term: $$= x(3x + 1) + 2(3x + 1)$$ $$= 3x^2 + x + 6x + 2$$ Combine like terms: $$= 3x^2 + 7x + 2$$ **c.** $(x - 4)^2$ Rewrite as $(x - 4)(x - 4)$ and distribute: $$= x(x - 4) - 4(x - 4)$$ $$= x^2 - 4x - 4x + 16$$ Combine like terms: $$= x^2 - 8x + 16$$ **Final answers:** a. $x^2 + 6x + 8$ b. $3x^2 + 7x + 2$ c. $x^2 - 8x + 16$