The given expression is:

\[
\frac{2x}{\frac{x^2 - y^2}{y} + 2y}
\]

We will simplify it step by step.

### Step 1: Simplify the denominator
The denominator is:

\[
\frac{x^2 - y^2}{y} + 2y
\]

Notice that $x^2 - y^2$ is a difference of squares:

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

So, the denominator becomes:

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

### Step 2: Combine the terms
Now distribute the denominator:

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

Thus, we have:

\[
\frac{2x}{\frac{x^2 - y^2}{y} + 2y} = \frac{2x}{\frac{x^2 - y^2}{y} + 2y}
\]

To combine these two terms, find the common denominator:

\[
\frac{x^2 - y^2}{y} + 2y = \frac{x^2 - y^2 + 2y^2}{y} = \frac{x^2 + y^2}{y}
\]

### Step 3: Final simplification
Now, the expression becomes:

\[
\frac{2x}{\frac{x^2 + y^2}{y}} = 2x \cdot \frac{y}{x^2 + y^2} = \frac{2xy}{x^2 + y^2}
\]

### Final Answer:
\[
\boxed{\frac{2xy}{x^2 + y^2}}
\]

Would you like more details on any step?

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Here are 5 related questions to expand your understanding:

1. How can you use this technique to simplify other rational expressions?
2. What is the significance of the difference of squares in algebraic simplifications?
3. How can you verify the result by substituting specific values for $x$ and $y$?
4. What other forms can the final expression take if simplified differently?
5. How does factoring help in simplifying complex algebraic fractions?

**Tip:** Always look for common factorizations like difference of squares to simplify complex expressions quickly.

$$ \frac{2x}{\frac{x^2 - y^2}{y} + 2y} $$

Step-by-step guide to simplify the expression $$ \frac{2x}{\frac{x^2 - y^2}{y} + 2y} $$ using algebraic techniques like the difference of squares. Suitable for high school students learning advanced algebraic simplifications.

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