The expression you provided is:

$\frac{3}{x + 7} + \frac{4}{x - 8}$

To simplify this expression, we need to find a common denominator.

Step 1: Identify the common denominator

The common denominator between $x + 7$ and $x - 8$ is simply the product of the two:

$(x + 7)(x - 8)$

Step 2: Rewrite each fraction with the common denominator

$\frac{3}{x + 7} = \frac{3(x - 8)}{(x + 7)(x - 8)}$ $\frac{4}{x - 8} = \frac{4(x + 7)}{(x - 8)(x + 7)}$

Step 3: Combine the two fractions

$\frac{3(x - 8) + 4(x + 7)}{(x + 7)(x - 8)}$

Step 4: Distribute and simplify the numerator

Distribute the constants inside the parentheses:

$3(x - 8) = 3x - 24$ $4(x + 7) = 4x + 28$

Now, combine the expressions:

$\frac{3x - 24 + 4x + 28}{(x + 7)(x - 8)} = \frac{7x + 4}{(x + 7)(x - 8)}$

Final Simplified Expression:

$\frac{7x + 4}{(x + 7)(x - 8)}$

Would you like more details or have any questions? Here are some related questions:

1. What if the denominators had been quadratic expressions?
2. How do we simplify more complex rational expressions?
3. Can you explain how to factor polynomials in the denominator to find a common denominator?
4. What happens if $x = -7$ or $x = 8$ in this expression?
5. How can we apply this to solving equations involving rational expressions?

Tip: When adding or subtracting fractions, always make sure to use a common denominator before combining the terms.