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.