We are given the equation $a^2 - b^2 = 2ab$, and we want to find the value of $\frac{a+b}{a-b}$.

Step 1: Use the difference of squares identity

The left-hand side of the equation is a difference of squares, which can be factored as: $a^2 - b^2 = (a + b)(a - b)$ So, the equation becomes: $(a + b)(a - b) = 2ab$

Step 2: Solve for $\frac{a+b}{a-b}$

Now divide both sides of the equation by $(a - b)$ (assuming $a \neq b$) to isolate $a + b$: $a + b = \frac{2ab}{a - b}$ Finally, divide both sides by $a - b$ to find $\frac{a+b}{a-b}$: $\frac{a+b}{a-b} = \frac{2ab}{(a-b)^2}$

Thus, the value of $\frac{a+b}{a-b}$ is $\frac{2ab}{(a-b)^2}$.

Would you like further details or clarifications?

Here are some related questions to explore:

1. How does the solution change if $a = b$?
2. Can this problem be solved geometrically or graphically?
3. How would this equation change if $a^2 - b^2 = 3ab$ instead?
4. What would happen if $b = 0$ in this equation?
5. Can we express $a + b$ in terms of other variables from this equation?

Tip: When dealing with factorizations like difference of squares, always check if the equation can be simplified further!