To simplify the expression $\frac{5}{x+4} + \frac{7}{x+3}$, we need to find a common denominator.

Steps:

1. Identify the denominators: The denominators are $x+4$ and $x+3$.
2. Find the common denominator: The common denominator is the product of the two distinct denominators, which is $(x+4)(x+3)$.
3. Rewrite each fraction with the common denominator:
   • $\frac{5}{x+4}$ becomes $\frac{5(x+3)}{(x+4)(x+3)}$.
   • $\frac{7}{x+3}$ becomes $\frac{7(x+4)}{(x+4)(x+3)}$.
4. Add the fractions: Now that both fractions have the same denominator, we can add the numerators:

$\frac{5(x+3)}{(x+4)(x+3)} + \frac{7(x+4)}{(x+4)(x+3)} = \frac{5(x+3) + 7(x+4)}{(x+4)(x+3)}$

5. Simplify the numerator: $5(x+3) + 7(x+4) = 5x + 15 + 7x + 28 = 12x + 43$

Thus, the expression simplifies to:

$\frac{12x + 43}{(x+4)(x+3)}$

Final answer:

$\frac{12x + 43}{(x+4)(x+3)}$

Would you like further details on this solution or have any questions? Here are some related questions that might interest you:

1. How do you simplify expressions with more than two fractions?
2. What are the rules for adding rational expressions?
3. How do you factor the denominator in rational expressions?
4. Can this expression be simplified further in any way?
5. How do you find the domain of rational expressions?

Tip: When adding fractions with different denominators, always factor the denominators and find the least common denominator (LCD) to make addition easier.