To identify which expression simplifies to [tex]\(\frac{m-2}{m-1}\)[/tex], we need to simplify the given fractions and compare the results.
Let's simplify each fraction step-by-step:
### Option A: [tex]\(\frac{m^2-3m+2}{m^2-m}\)[/tex]
1. Factorize the numerator and the denominator:
- [tex]\(m^2-3m+2 = (m-1)(m-2)\)[/tex]
- [tex]\(m^2-m = m(m-1)\)[/tex]
2. Write the fraction with the factored terms:
[tex]\[
\frac{(m-1)(m-2)}{m(m-1)}
\][/tex]
3. Cancel the common terms [tex]\((m-1)\)[/tex]:
[tex]\[
\frac{m-2}{m}
\][/tex]
This is not equal to [tex]\(\frac{m-2}{m-1}\)[/tex].
### Option B: [tex]\(\frac{m^2-2m+1}{m-1}\)[/tex]
1. Factorize the numerator:
- [tex]\(m^2-2m+1 = (m-1)^2\)[/tex]
2. Write the fraction with the factored terms:
[tex]\[
\frac{(m-1)^2}{m-1}
\][/tex]
3. Cancel the common term [tex]\(m-1\)[/tex]:
[tex]\[
m-1
\][/tex]
This is not equal to [tex]\(\frac{m-2}{m-1}\)[/tex].
### Option C: [tex]\(\frac{m^2-m-2}{m^2-1}\)[/tex]
1. Factorize the numerator and the denominator:
- [tex]\(m^2-m-2 = (m-2)(m+1)\)[/tex]
- [tex]\(m^2-1 = (m-1)(m+1)\)[/tex]
2. Write the fraction with the factored terms:
[tex]\[
\frac{(m-2)(m+1)}{(m-1)(m+1)}
\][/tex]
3. Cancel the common term [tex]\((m+1)\)[/tex]:
[tex]\[
\frac{m-2}{m-1}
\][/tex]
This matches the target fraction [tex]\(\frac{m-2}{m-1}\)[/tex].
### Option D: [tex]\(\frac{2m^2-4m}{2(m-2)}\)[/tex]
1. Factorize the numerator and simplify:
- [tex]\(2m^2-4m = 2m(m-2)\)[/tex]
2. Write the fraction with the factored terms:
[tex]\[
\frac{2m(m-2)}{2(m-2)}
\][/tex]
3. Cancel the common terms [tex]\(2(m-2)\)[/tex]:
[tex]\[
m
\][/tex]
This is not equal to [tex]\(\frac{m-2}{m-1}\)[/tex].
### Conclusion:
The correct expression that simplifies to [tex]\(\frac{m-2}{m-1}\)[/tex] is:
[tex]\[
\boxed{C. \frac{m^2-m-2}{m^2-1}}
\][/tex]
