The problem asks to divide the polynomial $108m^3n - 122mn + 48mn^2$ by $2mn$.

1. Problem Description

The problem asks to divide the polynomial $108m^3n - 122mn + 48mn^2$ by $2mn$.

2. Solution Steps

We divide each term of the polynomial by $2mn$:

$\frac{108m^3n - 122mn + 48mn^2}{2mn} = \frac{108m^3n}{2mn} - \frac{122mn}{2mn} + \frac{48mn^2}{2mn}$

Now, we simplify each term:

$\frac{108m^3n}{2mn} = \frac{108}{2} \cdot \frac{m^3}{m} \cdot \frac{n}{n} = 54m^2$

$\frac{122mn}{2mn} = \frac{122}{2} \cdot \frac{m}{m} \cdot \frac{n}{n} = 61$

$\frac{48mn^2}{2mn} = \frac{48}{2} \cdot \frac{m}{m} \cdot \frac{n^2}{n} = 24n$

Therefore,

$\frac{108m^3n - 122mn + 48mn^2}{2mn} = 54m^2 - 61 + 24n$.

3. Final Answer

$54m^2 - 61 + 24n$