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, we factor the denominator of the left side:

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

So we have

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

Next, we combine the fractions on the right side:

$\frac{4}{x+5} - \frac{2}{2x-3} = \frac{4(2x-3) - 2(x+5)}{(x+5)(2x-3)} = \frac{8x - 12 - 2x - 10}{(x+5)(2x-3)} = \frac{6x - 22}{(x+5)(2x-3)}$.

Now we have

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

Since the denominators are the same, we can equate the numerators:

$6x + m = 6x - 22$.

Subtracting $6x$ from both sides gives

$m = -22$.

3. Final Answer

$m = -22$