The problem requires performing the indicated addition and reducing the result if possible. Here's how to approach it:

Given Expression

$\frac{x - 3}{x + 4} + \frac{x + 11}{x + 4}$

Since the denominators are the same ($x + 4$), we can combine the numerators directly.

Step-by-Step Solution

1. Combine the Numerators: $\frac{(x - 3) + (x + 11)}{x + 4}$

2. Simplify the Numerator: $(x - 3) + (x + 11) = x + x - 3 + 11 = 2x + 8$

3. Write the Combined Fraction: $\frac{2x + 8}{x + 4}$

4. Factor the Numerator (if possible): The numerator $2x + 8$ can be factored as: $2x + 8 = 2(x + 4)$

5. Reduce the Fraction: $\frac{2(x + 4)}{x + 4}$

   Since $x + 4 \neq 0$ (as stated in the problem), we can cancel the $x + 4$ terms: $2$

Final Answer

$2$

Would you like further details or have any questions?

Follow-up Questions

1. How would the problem change if the denominators were different?
2. What are the restrictions on the values of $x$ in this problem?
3. Can you explain why we are allowed to cancel out the $x + 4$ in the final fraction?
4. How would you handle this problem if you were given subtraction instead of addition?
5. Can this technique be used to simplify more complex rational expressions?

Tip

Always check for common factors in the numerator and denominator before simplifying a rational expression.