The expression in the image is:

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

To combine these fractions, we'll need to find a common denominator.

Step-by-Step Solution:

1. Identify the denominators: The denominators are $x + 7$ and $x - 8$.

2. Find the least common denominator (LCD): Since the denominators are distinct and cannot be simplified further, the LCD is simply the product of the two, $(x + 7)(x - 8)$.

3. Rewrite each fraction with the LCD:

   • For $\frac{3}{x + 7}$, multiply the numerator and the denominator by $x - 8$ to get $\frac{3(x - 8)}{(x + 7)(x - 8)}$.
   • For $\frac{4}{x - 8}$, multiply the numerator and the denominator by $x + 7$ to get $\frac{4(x + 7)}{(x + 7)(x - 8)}$.

4. Combine the fractions: $\frac{3(x - 8)}{(x + 7)(x - 8)} + \frac{4(x + 7)}{(x + 7)(x - 8)} = \frac{3(x - 8) + 4(x + 7)}{(x + 7)(x - 8)}$

5. Simplify the numerator:

   • Distribute the constants: $3(x - 8) = 3x - 24$ $4(x + 7) = 4x + 28$
   • Add the results: $3x - 24 + 4x + 28 = 7x + 4$

6. Final simplified expression: $\frac{7x + 4}{(x + 7)(x - 8)}$

This is the simplified form of the given expression.

Would you like further details or have any questions?

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Here are 8 related questions:

1. How do you find the least common denominator when the denominators are more complex?
2. What are the general rules for distributing multiplication over addition or subtraction?
3. How do you solve an equation if the expression equals a specific value, like zero?
4. Can you further factorize the numerator in this case?
5. How would you handle similar problems with three fractions instead of two?
6. What happens if you substitute specific values for $x$ in the simplified expression?
7. What is the significance of checking for restrictions in the domain when dealing with denominators?
8. How would this process change if one of the denominators were a quadratic expression?

Tip: Always check your final expression for potential domain restrictions—values of $x$ that would make any denominator zero—before considering the solution complete.