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 find the expression that simplifies to \( \frac{m-2}{m-1} \), first simplify each option. For option A: \( \frac{m^{2}-3m+2}{m^{2}-m} \) Factoring both the numerator and the denominator gives \( \frac{(m-1)(m-2)}{m(m-1)} = \frac{m-2}{m} \) (not correct). For option B: \( \frac{m^{2}-2m+1}{m-1} \) Factoring the numerator gives \( \frac{(m-1)(m-1)}{m-1} = m-1 \) (not correct). For option C: \( \frac{m^{2}-m-2}{m^{2}-1} \) Factoring the numerator gives \( \frac{(m-2)(m+1)}{(m-1)(m+1)} = \frac{m-2}{m-1} \) (this works!). For option D: \( \frac{2m^{2}-4m}{2(m-2)} \) Simplifying gives \( \frac{2m(m-2)}{2(m-2)} = m \) (not correct). So, the correct answer is **C**. Now isn't it fun how mathematics unfolds? Understanding how to factor and simplify expressions opens the door to quicker problem-solving! Speaking of problem-solving, always double-check your factorizations as small mistakes can lead you astray. For instance, missing a common factor can change everything, just like a magician performing a trick! Keep practicing, and it'll become second nature.