To determine which equation has infinitely many solutions, we need to check each option to see if they resolve to an identity (a statement that is true for all values of \( x \)).
Option A:
\[ \frac{8x + 3}{2} = 4x + 5 \] Multiply both sides by 2: \[ 8x + 3 = 8x + 10 \] Subtract \( 8x \) from both sides: \[ 3 = 10 \] This is not true; thus, option A does not have infinitely many solutions.
Option B:
\[ \frac{6x + 5}{4} = 2x + 2 \] Multiply both sides by 4: \[ 6x + 5 = 8x + 8 \] Subtract \( 6x \) from both sides: \[ 5 = 2x + 8 \] Subtract 8 from both sides: \[ -3 = 2x \] Divide by 2: \[ x = -\frac{3}{2} \] This has a unique solution; thus, option B does not have infinitely many solutions.
Option C:
\[ \frac{12x + 32}{4} = 3x + 8 \] Multiply both sides by 4: \[ 12x + 32 = 12x + 32 \] This equation is true for all \( x \); thus, option C has infinitely many solutions.
Option D:
\[ \frac{8x - 5}{2} = 2x - 6 \] Multiply both sides by 2: \[ 8x - 5 = 4x - 12 \] Subtract \( 4x \) from both sides: \[ 4x - 5 = -12 \] Add 5 to both sides: \[ 4x = -7 \] Divide by 4: \[ x = -\frac{7}{4} \] This has a unique solution; thus, option D does not have infinitely many solutions.
Conclusion
The only equation that has infinitely many solutions is Option C: \[ \frac{12x + 32}{4} = 3x + 8 \]