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The given expression is:

\[
\frac{\frac{2}{x-2} + \frac{1}{x+2}}{\frac{6}{x^2 - 4}}
\]

### Step-by-step simplification:

#### 1. Simplify the denominator:
The denominator is \(\frac{6}{x^2 - 4}\). We can factor \(x^2 - 4\) as:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

So the denominator becomes:

\[
\frac{6}{(x-2)(x+2)}
\]

#### 2. Simplify the numerator:
The numerator is the sum of two fractions:

\[
\frac{2}{x-2} + \frac{1}{x+2}
\]

To add these fractions, we need a common denominator. The common denominator is \((x - 2)(x + 2)\). So we rewrite both fractions with this common denominator:

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

Now, combine the numerators:

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

Simplify the numerator:

\[
2(x+2) + (x-2) = 2x + 4 + x - 2 = 3x + 2
\]

So the entire numerator is:

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

#### 3. Combine the entire expression:
Now, we have:

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

This is a complex fraction, and we can simplify by multiplying by the reciprocal of the denominator:

\[
\frac{3x + 2}{(x-2)(x+2)} \times \frac{(x-2)(x+2)}{6}
\]

Cancel out \((x-2)(x+2)\) from the numerator and denominator:

\[
\frac{3x + 2}{6}
\]

Thus, the simplified expression is:

\[
\frac{3x + 2}{6}
\]

Let me know if you'd like more details or if you have any questions!

### Related Questions:
1. How do you add fractions with different denominators?
2. What is the factored form of \(x^2 - 4\)?
3. How do you simplify complex fractions?
4. Why is it important to cancel common factors in fractions?
5. What are the rules for simplifying algebraic expressions?

### Tip:
When simplifying complex fractions, always look for common factors that can be canceled out to make the process easier.

Simplify the expression: (2 / (x - 2) + 1 / (x + 2)) / (6 / (x^2 - 4))

Learn how to simplify the complex fraction (2/(x - 2) + 1/(x + 2)) ÷ (6/(x^2 - 4)) with a step-by-step solution. This problem involves algebraic techniques such as factoring quadratics and adding fractions with different denominators. Suitable for students in Grades 10-12.

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