The problem asks to simplify the expression $\frac{x+7}{x^2+x-56} - \frac{x+8}{x^2-49}$.

1. Problem Description

The problem asks to simplify the expression $\frac{x+7}{x^2+x-56} - \frac{x+8}{x^2-49}$.

2. Solution Steps

First, factor the denominators of both fractions.

$x^2+x-56 = (x+8)(x-7)$

$x^2-49 = (x+7)(x-7)$

So the expression becomes:

$\frac{x+7}{(x+8)(x-7)} - \frac{x+8}{(x+7)(x-7)}$

The least common denominator is $(x+8)(x-7)(x+7)$.

We rewrite each fraction with the common denominator:

$\frac{(x+7)(x+7)}{(x+8)(x-7)(x+7)} - \frac{(x+8)(x+8)}{(x+7)(x-7)(x+8)}$

Now, subtract the numerators:

$\frac{(x+7)(x+7) - (x+8)(x+8)}{(x+8)(x-7)(x+7)}$

Expanding the numerators gives:

$\frac{x^2 + 14x + 49 - (x^2 + 16x + 64)}{(x+8)(x-7)(x+7)}$

$\frac{x^2 + 14x + 49 - x^2 - 16x - 64}{(x+8)(x-7)(x+7)}$

Simplify the numerator:

$\frac{-2x - 15}{(x+8)(x-7)(x+7)}$

3. Final Answer

$\frac{-2x-15}{(x+8)(x-7)(x+7)}$