We are given the equation $\frac{6x+m}{2x^2+7x-15} = \frac{4}{x+5} - \frac{2}{2x-3}$, and we need to find the value of $m$.

1. Problem Description

We are given the equation $\frac{6x+m}{2x^2+7x-15} = \frac{4}{x+5} - \frac{2}{2x-3}$, and we need to find the value of $m$.

2. Solution Steps

First, factor the quadratic expression in the denominator on the left side:

$2x^2 + 7x - 15 = (x+5)(2x-3)$

So the given equation becomes:

$\frac{6x+m}{(x+5)(2x-3)} = \frac{4}{x+5} - \frac{2}{2x-3}$

Now, find a common denominator for the right side:

$\frac{4}{x+5} - \frac{2}{2x-3} = \frac{4(2x-3) - 2(x+5)}{(x+5)(2x-3)}$

Simplify the numerator:

$4(2x-3) - 2(x+5) = 8x - 12 - 2x - 10 = 6x - 22$

So the equation becomes:

$\frac{6x+m}{(x+5)(2x-3)} = \frac{6x-22}{(x+5)(2x-3)}$

Since the denominators are the same, the numerators must be equal:

$6x + m = 6x - 22$

Subtract $6x$ from both sides:

$m = -22$

3. Final Answer

The value of $m$ is $-22$.