The problem asks us to simplify the following expression: $\frac{\frac{3m^2 + 2m - 1}{m^2 - 1}}{\frac{2m - 1}{m^2 - 2m + 1}}$

1. Problem Description

The problem asks us to simplify the following expression:

$\frac{\frac{3m^2 + 2m - 1}{m^2 - 1}}{\frac{2m - 1}{m^2 - 2m + 1}}$

2. Solution Steps

We can rewrite the expression as a division of two fractions:

$\frac{3m^2 + 2m - 1}{m^2 - 1} \div \frac{2m - 1}{m^2 - 2m + 1}$

To divide fractions, we multiply by the reciprocal of the second fraction:

$\frac{3m^2 + 2m - 1}{m^2 - 1} \cdot \frac{m^2 - 2m + 1}{2m - 1}$

Now, we factor the polynomials:

$3m^2 + 2m - 1 = (3m - 1)(m + 1)$

$m^2 - 1 = (m - 1)(m + 1)$

$m^2 - 2m + 1 = (m - 1)(m - 1) = (m - 1)^2$

$2m - 1 = 2m - 1$

Substitute the factored expressions into the equation:

$\frac{(3m - 1)(m + 1)}{(m - 1)(m + 1)} \cdot \frac{(m - 1)^2}{2m - 1}$

Now, cancel common factors:

$\frac{(3m - 1)}{(m - 1)} \cdot \frac{(m - 1)^2}{2m - 1} = \frac{(3m - 1)(m - 1)(m - 1)}{(m - 1)(2m - 1)}$

Cancel $(m - 1)$:

$\frac{(3m - 1)(m - 1)}{2m - 1}$

Therefore, we have:

$\frac{(3m - 1)(m - 1)}{2m - 1}$

3. Final Answer

$\frac{(3m-1)(m-1)}{2m-1}$