The expression given in the image is:

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

Let's break this down step by step.

Step 1: Simplify the numerator

The numerator is: $x + \frac{2x}{x - 2}$ To simplify this, we need a common denominator: $\frac{x(x - 2)}{x - 2} + \frac{2x}{x - 2} = \frac{x(x - 2) + 2x}{x - 2}$ Expanding $x(x - 2)$: $x(x - 2) = x^2 - 2x$ Thus, the numerator becomes: $\frac{x^2 - 2x + 2x}{x - 2} = \frac{x^2}{x - 2}$

Step 2: Simplify the denominator

The denominator is: $1 + \frac{4}{(x + 2)(x - 2)}$ This can be rewritten with a common denominator: $\frac{(x + 2)(x - 2)}{(x + 2)(x - 2)} + \frac{4}{(x + 2)(x - 2)} = \frac{(x + 2)(x - 2) + 4}{(x + 2)(x - 2)}$ Expanding $(x + 2)(x - 2)$: $(x + 2)(x - 2) = x^2 - 4$ Thus, the denominator becomes: $\frac{x^2 - 4 + 4}{x^2 - 4} = \frac{x^2}{x^2 - 4}$

Step 3: Combine the numerator and denominator

We now have: $\frac{\frac{x^2}{x - 2}}{\frac{x^2}{x^2 - 4}} = \frac{x^2}{x - 2} \times \frac{x^2 - 4}{x^2}$ Cancel out $x^2$ in the numerator and denominator: $\frac{x^2 - 4}{x - 2}$ Now, recall that: $x^2 - 4 = (x - 2)(x + 2)$ So the expression becomes: $\frac{(x - 2)(x + 2)}{x - 2}$ Finally, cancel out $x - 2$ (as long as $x \neq 2$): $x + 2$

Thus, the simplified expression is:

$x + 2$

Would you like further explanation or details? Here are some follow-up questions:

1. How do common denominators work when simplifying complex fractions?
2. What conditions must be considered when canceling out terms like $(x - 2)$?
3. How does factoring help in simplifying rational expressions?
4. Can you solve this type of problem using another method?
5. What are the restrictions on the variable $x$ in the original expression?

Tip: Always check for restrictions on the variable when simplifying rational expressions (like $x \neq 2$ and $x \neq -2$ here).