Select the expressions that are equivalent to \( 6(4 f-2)-5 \). \[ \begin{array}{c}(-4 f+8 f-2) 6-5 \\ 6(-2+4 f)-5 \\ -17 f+24 \\ 6(f+3 f-2)-5\end{array} \]

To solve the problem, we first need to simplify the expression \( 6(4f - 2) - 5 \). 1. Expand \( 6(4f - 2) \): \[ 6 \cdot 4f - 6 \cdot 2 = 24f - 12 \] 2. Now subtract 5: \[ 24f - 12 - 5 = 24f - 17 \] Now we need to check which of the given expressions is equivalent to \( 24f - 17 \). 1. For the first expression \( (-4f + 8f - 2)6 - 5 \): - Simplifying, we get \( (8f - 4f - 2)6 - 5 = (4f - 2)6 - 5 = 24f - 12 - 5 = 24f - 17\). 2. For the second expression \( 6(-2 + 4f) - 5 \): - Simplifying, we get \( 6 \cdot (-2) + 6 \cdot 4f - 5 = -12 + 24f - 5 = 24f - 17\). 3. For the third expression \( -17f + 24 \): - This is not equivalent to \( 24f - 17\). 4. For the fourth expression \( 6(f + 3f - 2) - 5 \): - Simplifying, \( 6(4f - 2) - 5 = 24f - 12 - 5 = 24f - 17 \). The expressions equivalent to \( 6(4f - 2) - 5 \) are the first, second, and fourth expressions. **Final Result:** - **First Expression:** Yes, \( 6(4f - 2) - 5 \equiv (-4f + 8f - 2)6 - 5 \). - **Second Expression:** Yes, \( 6(4f - 2) - 5 \equiv 6(-2 + 4f) - 5 \). - **Third Expression:** No, because it simplifies to \(-17f + 24\). - **Fourth Expression:** Yes, \( 6(4f - 2) - 5 \equiv 6(f + 3f - 2) - 5 \). Thus, the expressions that are equivalent to \( 6(4 f-2)-5 \) are: - \( (-4 f + 8 f - 2) 6 - 5 \) - \( 6(-2 + 4 f) - 5 \) - \( 6(f + 3 f - 2) - 5 \)