We are asked to simplify the following expression: $\frac{3m}{4m+8} + \frac{4m-n}{3m+6} - \frac{2n}{m^2+2m}$.

1. Problem Description

We are asked to simplify the following expression:

$\frac{3m}{4m+8} + \frac{4m-n}{3m+6} - \frac{2n}{m^2+2m}$.

2. Solution Steps

First, we factor the denominators:

$4m+8 = 4(m+2)$

$3m+6 = 3(m+2)$

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

The given expression becomes:

$\frac{3m}{4(m+2)} + \frac{4m-n}{3(m+2)} - \frac{2n}{m(m+2)}$

To combine the fractions, we need a common denominator. The least common denominator is $12m(m+2)$.

Now, we rewrite each fraction with the common denominator:

$\frac{3m}{4(m+2)} \cdot \frac{3m}{3m} = \frac{9m^2}{12m(m+2)}$

$\frac{4m-n}{3(m+2)} \cdot \frac{4m}{4m} = \frac{4m(4m-n)}{12m(m+2)} = \frac{16m^2-4mn}{12m(m+2)}$

$\frac{2n}{m(m+2)} \cdot \frac{12}{12} = \frac{24n}{12m(m+2)}$

Thus, the expression becomes:

$\frac{9m^2}{12m(m+2)} + \frac{16m^2-4mn}{12m(m+2)} - \frac{24n}{12m(m+2)}$

Combining the numerators, we get:

$\frac{9m^2 + 16m^2 - 4mn - 24n}{12m(m+2)} = \frac{25m^2 - 4mn - 24n}{12m(m+2)}$

The numerator cannot be factored further.

Therefore, the simplified expression is:

$\frac{25m^2 - 4mn - 24n}{12m(m+2)}$.

3. Final Answer

$\frac{25m^2 - 4mn - 24n}{12m(m+2)}$