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 $x^2 - 49$. This is a difference of squares, so $x^2 - 49 = (x-7)(x+7)$.

Then, we rewrite the expression:

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

To add the two fractions, we need a common denominator. The common denominator is $(x-7)(x+7)$.

Multiply the first fraction by $\frac{x-7}{x-7}$ to get the common denominator:

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

Now, the expression becomes:

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

Combine the two fractions:

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

Simplify the numerator:

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

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

The denominator is $(x-7)(x+7) = x^2 - 49$. So, we can write the simplified expression as:

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

3. Final Answer

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