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.

Learn how to simplify the rational expression 3/(x + 7) + 4/(x - 8) step-by-step with a detailed solution. This problem involves rational expressions and finding common denominators. Suitable for high school students.

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