1. **State the problem:** We want to find which expressions are equivalent to $$4m + 4n + 10$$. 2. **Recall the distributive property:** $$a(b + c) = ab + ac$$. This helps us expand or factor expressions. 3. **Check each expression:** - Expression 1: $$4(m + n + \frac{5}{2}) = 4m + 4n + 4 \times \frac{5}{2} = 4m + 4n + 10$$. Equivalent. - Expression 2: $$4(m + n) + 10 = 4m + 4n + 10$$. Equivalent. - Expression 3: $$4m + 2(2n + 10) = 4m + 4n + 20$$. Not equivalent. - Expression 4: $$4(m + n) + \frac{5}{2} = 4m + 4n + 2.5$$. Not equivalent. - Expression 5: $$4m + 2(2n + 5) = 4m + 4n + 10$$. Equivalent. - Expression 6: $$10(6m + 6n + 1) = 60m + 60n + 10$$. Not equivalent. **Final answer:** Expressions 1, 2, and 5 are equivalent to $$4m + 4n + 10$$.