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It seems there was an issue processing the image through the current method. I will explain the steps involved in simplifying the given expression manually:

You provided the expression:
\[
\frac{x + \frac{2x}{x - 2}}{1 + \frac{4}{(x + 2)(x - 2)}}
\]

### Steps for simplification:

1. **Simplify the numerator**:
   The numerator is \(x + \frac{2x}{x - 2}\).
   - To combine the terms, we need a common denominator:
     \[
     x + \frac{2x}{x - 2} = \frac{x(x - 2) + 2x}{x - 2} = \frac{x^2 - 2x + 2x}{x - 2} = \frac{x^2}{x - 2}
     \]
   So, the numerator becomes \(\frac{x^2}{x - 2}\).

2. **Simplify the denominator**:
   The denominator is \(1 + \frac{4}{(x + 2)(x - 2)}\).
   - We rewrite this as:
     \[
     1 + \frac{4}{(x^2 - 4)} = \frac{(x^2 - 4) + 4}{x^2 - 4} = \frac{x^2}{x^2 - 4}
     \]
   So, the denominator becomes \(\frac{x^2}{x^2 - 4}\).

3. **Final expression**:
   Now, we divide the simplified numerator by the simplified denominator:
   \[
   \frac{\frac{x^2}{x - 2}}{\frac{x^2}{x^2 - 4}} = \frac{x^2}{x - 2} \times \frac{x^2 - 4}{x^2}
   \]
   - The \(x^2\) terms cancel out:
     \[
     \frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}
     \]
   - Finally, cancel out \(x - 2\):
     \[
     x + 2
     \]

### Final simplified expression:
\[
x + 2
\]

Let me know if you would like further explanation or if you have any questions.

Here are 5 related questions:
1. How do you simplify rational expressions with multiple terms?
2. What is the method to find a common denominator in algebraic fractions?
3. How do you handle expressions that involve factoring quadratics?
4. What is the concept behind canceling common factors in algebraic expressions?
5. Can you apply these steps to solve equations involving more variables?

**Tip**: When simplifying complex fractions, always factor expressions where possible and look for common factors to cancel.

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

Learn how to simplify the complex rational expression (x + 2x / (x - 2)) / (1 + 4 / ((x + 2)(x - 2))) with a detailed step-by-step solution. This guide covers key algebraic concepts such as combining terms, simplifying complex fractions, and factoring.

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