The problem asks us to add and simplify the expression $\frac{4}{x+7} + \frac{3}{x^2 - 49}$.

1. Problem Description

The problem asks us to add and simplify the expression $\frac{4}{x+7} + \frac{3}{x^2 - 49}$.

2. Solution Steps

First, we factor the denominator of the second term:

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

So the expression becomes:

$\frac{4}{x+7} + \frac{3}{(x+7)(x-7)}$

To add the two fractions, we need to find a common denominator, which is $(x+7)(x-7)$. We multiply the first fraction by $\frac{x-7}{x-7}$ to get a common denominator:

$\frac{4(x-7)}{(x+7)(x-7)} + \frac{3}{(x+7)(x-7)}$

Now we can add the numerators:

$\frac{4(x-7) + 3}{(x+7)(x-7)}$

Distribute the 4 in the numerator:

$\frac{4x - 28 + 3}{(x+7)(x-7)}$

Simplify the numerator:

$\frac{4x - 25}{(x+7)(x-7)}$

Since $x^2 - 49 = (x+7)(x-7)$, we can also write the result as:

$\frac{4x - 25}{x^2 - 49}$

3. Final Answer

$\frac{4x-25}{x^2-49}$