Select the correct answer. Which expression simplifies to \( \frac{m-2}{m-1} \) ? A. \( \frac{m^{2}-3 m+2}{m^{2}-m} \) B. \( \frac{m^{2}-2 m+1}{m-1} \) C. \( \frac{m^{2}-m-2}{m^{2}-1} \) D. \( \frac{2 m^{2}-4 m}{2(m-2)} \)

To determine which expression simplifies to \( \frac{m-2}{m-1} \), let's evaluate each option: **Option A:** \( \frac{m^{2}-3m+2}{m^{2}-m} \) - Factor the numerator: \( m^{2}-3m+2 = (m-1)(m-2) \) - Factor the denominator: \( m^{2}-m = m(m-1) \) - Simplify: \( \frac{(m-1)(m-2)}{m(m-1)} = \frac{m-2}{m} \) *This simplifies to \( \frac{m-2}{m} \), not \( \frac{m-2}{m-1} \).* **Option B:** \( \frac{m^{2}-2m+1}{m-1} \) - Factor the numerator: \( m^{2}-2m+1 = (m-1)^2 \) - Simplify: \( \frac{(m-1)^2}{m-1} = m-1 \) *This simplifies to \( m-1 \), not \( \frac{m-2}{m-1} \).* **Option C:** \( \frac{m^{2}-m-2}{m^{2}-1} \) - Factor the numerator: \( m^{2}-m-2 = (m-2)(m+1) \) - Factor the denominator: \( m^{2}-1 = (m-1)(m+1) \) - Simplify: \( \frac{(m-2)(m+1)}{(m-1)(m+1)} = \frac{m-2}{m-1} \) *This matches the target expression.* **Option D:** \( \frac{2m^{2}-4m}{2(m-2)} \) - Factor the numerator: \( 2m^{2}-4m = 2m(m-2) \) - Simplify: \( \frac{2m(m-2)}{2(m-2)} = m \) *This simplifies to \( m \), not \( \frac{m-2}{m-1} \).* **Conclusion:** The correct answer is **Option C**. **Answer:** C