1. **Stating the problem:** We need to expand and simplify the products of binomials given in expressions a) through f) and then match them with the provided expressions A) through F). 2. **Formula and rules:** When multiplying two binomials, use the distributive property (FOIL method): $$ (x + y)(a + b) = xa + xb + ya + yb $$ Multiply each term in the first parentheses by each term in the second parentheses. 3. **Step-by-step expansions:** a) $ (2a + 4)(2a - 4) = 2a \cdot 2a + 2a \cdot (-4) + 4 \cdot 2a + 4 \cdot (-4) = 4a^2 - 8a + 8a - 16 = 4a^2 - 16 $ b) $ (2a + 4)(2a + 4) = 2a \cdot 2a + 2a \cdot 4 + 4 \cdot 2a + 4 \cdot 4 = 4a^2 + 8a + 8a + 16 = 4a^2 + 16a + 16 $ c) $ (2a - 4)(2a - 4) = 2a \cdot 2a + 2a \cdot (-4) + (-4) \cdot 2a + (-4) \cdot (-4) = 4a^2 - 8a - 8a + 16 = 4a^2 - 16a + 16 $ d) $ (4a - 4)(a + 4) = 4a \cdot a + 4a \cdot 4 - 4 \cdot a - 4 \cdot 4 = 4a^2 + 16a - 4a - 16 = 4a^2 + 12a - 16 $ e) $ (4a - 8)(a + 2) = 4a \cdot a + 4a \cdot 2 - 8 \cdot a - 8 \cdot 2 = 4a^2 + 8a - 8a - 16 = 4a^2 - 16 $ f) $ (8a - 4)(0.5a + 4) = 8a \cdot 0.5a + 8a \cdot 4 - 4 \cdot 0.5a - 4 \cdot 4 = 4a^2 + 32a - 2a - 16 = 4a^2 + 30a - 16 $ 4. **Matching with given expressions:** - a) $4a^2 - 16$ matches D) - b) $4a^2 + 16a + 16$ matches C) - c) $4a^2 - 16a + 16$ matches E) - d) $4a^2 + 12a - 16$ matches B) - e) $4a^2 - 16$ matches D) - f) $4a^2 + 30a - 16$ matches A) **Final answers:** a) = D) b) = C) c) = E) d) = B) e) = D) f) = A)