The expression shown in the image is:

$\frac{1}{x+2y} - \frac{1}{x-2y} + \frac{2x}{4y^2 - x^2}$

Let's break down and simplify this expression step by step.

Step-by-Step Solution:

1. Combine the first two terms: $\frac{1}{x+2y} - \frac{1}{x-2y}$ We can combine these terms by finding a common denominator: $\frac{1}{x+2y} - \frac{1}{x-2y} = \frac{(x-2y) - (x+2y)}{(x+2y)(x-2y)}$ Simplifying the numerator: $(x - 2y) - (x + 2y) = x - 2y - x - 2y = -4y$ Thus: $\frac{1}{x+2y} - \frac{1}{x-2y} = \frac{-4y}{(x+2y)(x-2y)}$ Recall that: $(x+2y)(x-2y) = x^2 - (2y)^2 = x^2 - 4y^2$ Therefore: $\frac{1}{x+2y} - \frac{1}{x-2y} = \frac{-4y}{x^2 - 4y^2}$

2. Simplify the third term: $\frac{2x}{4y^2 - x^2}$ Notice that: $4y^2 - x^2 = -(x^2 - 4y^2)$ So: $\frac{2x}{4y^2 - x^2} = -\frac{2x}{x^2 - 4y^2}$

3. Combine all terms: Now, the entire expression becomes: $\frac{-4y}{x^2 - 4y^2} - \frac{2x}{x^2 - 4y^2}$ Since the denominators are the same, we can combine the numerators: $\frac{-4y - 2x}{x^2 - 4y^2}$ Factor out a common term: $\frac{-2(2y + x)}{x^2 - 4y^2}$

Final Simplified Expression:

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

Would you like more detailed explanations on any particular step, or have any other questions about this problem?

Here are 5 related questions to explore further:

1. How would you simplify an expression involving multiple fractions with different denominators?
2. Can you find an alternative method to combine fractions with complex terms?
3. What are the conditions under which the expression is undefined?
4. How does factoring help in simplifying rational expressions?
5. What applications do rational expressions like this have in calculus or real-world problems?

Tip: Always look for common factors when dealing with fractions involving polynomials to simplify expressions more easily!