Let's simplify the given expression step-by-step. The expression in the image is:

$\frac{a+b}{2(a-b)} - \frac{a-b}{2(a+b)} - \frac{2b^2}{b^2 - a^2}$

Step 1: Simplify Each Fraction

First, recognize that $b^2 - a^2$ can be factored as a difference of squares:

$b^2 - a^2 = (b-a)(b+a)$

Thus, the third term becomes:

$\frac{2b^2}{(b-a)(b+a)}$

Step 2: Find a Common Denominator

The denominators are $2(a-b)$, $2(a+b)$, and $(b-a)(b+a)$. To combine these fractions, we need a common denominator. Notice that:

• $a-b = -(b-a)$, so the expression $(a-b)(a+b) = -(b-a)(b+a)$

Therefore, the least common multiple (LCM) of $2(a-b)$, $2(a+b)$, and $(b-a)(b+a)$ is:

$2(a+b)(a-b)$

Step 3: Rewrite Each Fraction with the Common Denominator

Rewrite each term with this common denominator:

1. The first term:
   $\frac{a+b}{2(a-b)} = \frac{(a+b)(a+b)}{2(a-b)(a+b)}$

2. The second term:
   $\frac{a-b}{2(a+b)} = \frac{(a-b)(a-b)}{2(a-b)(a+b)}$

3. The third term:
   $\frac{2b^2}{(b-a)(b+a)} = \frac{-2b^2}{2(a-b)(a+b)}$ (since $b^2 - a^2 = -(a-b)(a+b)$)

Step 4: Combine the Terms

Now that all terms have the same denominator, we can combine them:

$\frac{(a+b)^2 - (a-b)^2 - 2b^2}{2(a-b)(a+b)}$

Step 5: Simplify the Numerator

Expand and simplify the numerator:

1. Expand $(a+b)^2$ and $(a-b)^2$:

   $(a+b)^2 = a^2 + 2ab + b^2$ $(a-b)^2 = a^2 - 2ab + b^2$

2. Subtract these two expansions and simplify:

   $(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) - 2b^2$ $a^2 + 2ab + b^2 - a^2 + 2ab - b^2 - 2b^2 = 4ab - 2b^2$

3. Factor out a 2 from the numerator:

   $2(2ab - b^2)$

Step 6: Write the Final Simplified Expression

Divide the numerator by the common denominator:

$\frac{2(2ab - b^2)}{2(a-b)(a+b)} = \frac{2ab - b^2}{(a-b)(a+b)}$

Conclusion

The simplified form of the expression is:

$\frac{2ab - b^2}{(a-b)(a+b)}$

Would you like more details on any of these steps, or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How do you simplify expressions involving fractions with different denominators?
2. What are the rules for factoring polynomials?
3. How can you factorize expressions involving differences of squares?
4. How do you find a common denominator for rational expressions?
5. What are some common techniques for simplifying algebraic fractions?

Tip: When dealing with algebraic fractions, always look for common factors or ways to factor expressions to simplify the problem.