1. **Stating the problem:** We need to expand and simplify each of the expressions a) to f) by multiplying the parentheses and then compare the results to find which expressions are equal. 2. **Formula and rules:** When multiplying two parentheses, multiply each term in the first parentheses by each term in the second parentheses. Use distributive property: $ (x + y)(z + w) = xz + xw + yz + yw $. 3. **Calculate each expression:** 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. **Compare results with given answers:** - a) $4a^2 - 16$ matches D) and also e) equals $4a^2 - 16$ which 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) and a). - f) $4a^2 + 30a - 16$ matches A). 5. **Conclusion:** Expressions a) and e) are equal and correspond to D). **Final answer:** - a) = e) = D) $4a^2 - 16$ - b) = C) $4a^2 + 16a + 16$ - c) = E) $4a^2 - 16a + 16$ - d) = B) $4a^2 + 12a - 16$ - f) = A) $4a^2 + 30a - 16$