The problem shown is:

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

To solve this, we need to add the two fractions by finding a common denominator.

Step 1: Find the common denominator

The denominators are $x+2$ and $x+4$, so the least common denominator (LCD) is $(x+2)(x+4)$.

Step 2: Rewrite the fractions with the common denominator

$\frac{2}{x+2} + \frac{3}{x+4} = \frac{2(x+4)}{(x+2)(x+4)} + \frac{3(x+2)}{(x+2)(x+4)}$

Step 3: Simplify the numerators

Expand the numerators:

$2(x+4) = 2x + 8$ $3(x+2) = 3x + 6$

Now, the expression becomes:

$\frac{2x + 8}{(x+2)(x+4)} + \frac{3x + 6}{(x+2)(x+4)}$

Step 4: Combine the fractions

Now that the denominators are the same, we can combine the numerators:

$\frac{(2x + 8) + (3x + 6)}{(x+2)(x+4)} = \frac{5x + 14}{(x+2)(x+4)}$

Final Answer:

$\frac{5x + 14}{(x+2)(x+4)}$

Would you like further details on any step or additional explanations? Here are some related questions to explore:

1. How do you find the least common denominator of two rational expressions?
2. What is the process of expanding binomials like $2(x+4)$?
3. How do you simplify complex fractions involving polynomials?
4. What is the importance of factoring in algebraic expressions?
5. How do restrictions on variables arise in rational expressions?

Tip: When adding fractions with different denominators, always find the least common denominator and rewrite both fractions before combining them.