The problem requires us to add or subtract the two rational expressions:

$\frac{9}{x + 3} + \frac{2}{x - 4}$

To solve this, we need to find a common denominator.

Step 1: Determine the least common denominator (LCD)

The denominators are $x + 3$ and $x - 4$. The LCD will be the product of these two expressions: $(x + 3)(x - 4)$

Step 2: Rewrite each fraction with the common denominator

We multiply the numerator and denominator of the first fraction by $x - 4$ and the second fraction by $x + 3$: $\frac{9}{x + 3} = \frac{9(x - 4)}{(x + 3)(x - 4)}$ $\frac{2}{x - 4} = \frac{2(x + 3)}{(x + 3)(x - 4)}$

Step 3: Combine the fractions

Now, we add the numerators together: $\frac{9(x - 4) + 2(x + 3)}{(x + 3)(x - 4)}$

Simplifying the numerator: $9(x - 4) = 9x - 36$ $2(x + 3) = 2x + 6$

So the combined numerator is: $9x - 36 + 2x + 6 = 11x - 30$

Step 4: Final expression

The resulting expression is: $\frac{11x - 30}{(x + 3)(x - 4)}$

So, the correct answer is: $\boxed{\frac{11x - 30}{(x + 3)(x - 4)}}$

Final Answer:

The option matching this result is the third one in the list.

Would you like further details or have any questions on this?

Here are 5 related questions you could explore:

1. How do you handle rational expressions with more complicated numerators or denominators?
2. What happens if you have to subtract the fractions instead of adding them?
3. How can you simplify the rational expression if one of the factors cancels out?
4. What is the significance of finding the least common denominator in adding or subtracting rational expressions?
5. Can you solve this problem using a different method, such as cross-multiplication?

Tip: Always check for any common factors that could simplify the expression once you've combined the fractions!