Which expression is equivalent to the expression below? \( \frac{\frac{6 c^{2}+3 c}{-4 c+2}}{\frac{2 c+1}{4 c-2}} \)

To simplify the expression \( \frac{\frac{6 c^{2}+3 c}{-4 c+2}}{\frac{2 c+1}{4 c-2}} \), we can start by simplifying the fractions individually. The numerator can be factored: \( 6c^2 + 3c = 3c(2c + 1) \). The denominator of the first fraction can also be factored: \( -4c + 2 = -2(2c - 1) \). Now, substituting these factored forms into the expression gives: \( \frac{\frac{3c(2c+1)}{-2(2c-1)}}{\frac{2c+1}{4c-2}} \). Now, we can rewrite the expression as: \( \frac{3c(2c+1)}{-2(2c-1)} \cdot \frac{4c-2}{2c+1} \). Note that \( 4c - 2 = 2(2c - 1) \), so substituting that in we get: \( \frac{3c(2c+1)}{-2(2c-1)} \cdot \frac{2(2c-1)}{2c+1} \). Cancelling \( 2c + 1 \) in the numerator and denominator (assuming \( 2c + 1 \neq 0 \)), we have: \( \frac{3c}{-2} \cdot 2 = \frac{-3c}{2} \). So the expression simplifies to: \(\frac{-3c}{2}\).