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To simplify the expression:

\[
\frac{3}{x+2} + \frac{5}{x-2}
\]

we need a common denominator. Here’s the step-by-step solution:

### Step 1: Find the Common Denominator
The common denominator for \( \frac{3}{x+2} \) and \( \frac{5}{x-2} \) is \( (x+2)(x-2) \).

### Step 2: Rewrite Each Fraction with the Common Denominator
Rewrite each fraction so they both have \( (x+2)(x-2) \) as the denominator:

\[
\frac{3}{x+2} = \frac{3(x-2)}{(x+2)(x-2)}
\]
\[
\frac{5}{x-2} = \frac{5(x+2)}{(x+2)(x-2)}
\]

### Step 3: Combine the Fractions
Now, combine the fractions:

\[
\frac{3(x-2) + 5(x+2)}{(x+2)(x-2)}
\]

### Step 4: Expand the Numerator
Expand the terms in the numerator:

\[
= \frac{3x - 6 + 5x + 10}{(x+2)(x-2)}
\]

### Step 5: Simplify the Numerator
Combine like terms in the numerator:

\[
= \frac{(3x + 5x) + (-6 + 10)}{(x+2)(x-2)}
= \frac{8x + 4}{(x+2)(x-2)}
\]

### Step 6: Factor Out Common Terms (if possible)
In this case, we can factor out a 4 in the numerator:

\[
= \frac{4(2x + 1)}{(x+2)(x-2)}
\]

So, the simplified form of the expression is:

\[
\frac{4(2x + 1)}{(x+2)(x-2)}
\]

---

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. How do you simplify expressions with different denominators?
2. What is the importance of factoring in simplifying rational expressions?
3. How can you check if two fractions are equivalent?
4. Why is it important to simplify expressions in algebra?
5. How would this process change if there were additional terms in the numerator or denominator?

### Tip:
When simplifying fractions with algebraic denominators, always look for opportunities to factor both the numerator and denominator fully—it often leads to further simplification.

Simplify the following expression: 3/(x+2) + 5/(x-2)

Learn how to simplify the rational expression 3/(x+2) + 5/(x-2) with a step-by-step solution. This problem explores key algebraic concepts like finding a common denominator and combining fractions. Suitable for students in Grades 9-10.

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