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We are asked to perform the indicated operations for the given expression:

\[
\frac{2ab}{a^2 - b^2} - \frac{b}{a - b} + \frac{6}{2}
\]

### Step-by-Step Solution:

1. **Simplify the terms:**

   - The first term \(\frac{2ab}{a^2 - b^2}\): Notice that \(a^2 - b^2\) is a difference of squares and can be factored:
     \[
     a^2 - b^2 = (a - b)(a + b)
     \]
     So the first term becomes:
     \[
     \frac{2ab}{(a - b)(a + b)}
     \]

   - The second term \(\frac{b}{a - b}\) remains unchanged.

   - The third term \(\frac{6}{2}\) simplifies to \(3\).

2. **Common Denominator:**

   To combine the first two fractions, find a common denominator. The first term has a denominator of \((a - b)(a + b)\), while the second term has a denominator of \(a - b\). The least common denominator (LCD) is \((a - b)(a + b)\).

   Rewrite the second term to have this common denominator:
   \[
   \frac{b}{a - b} = \frac{b(a + b)}{(a - b)(a + b)}
   \]
   
3. **Combine the first two terms:**

   Now, subtract the two fractions:
   \[
   \frac{2ab}{(a - b)(a + b)} - \frac{b(a + b)}{(a - b)(a + b)} = \frac{2ab - b(a + b)}{(a - b)(a + b)}
   \]
   Expand the numerator of the second term:
   \[
   2ab - b(a + b) = 2ab - (ba + b^2) = 2ab - ba - b^2 = ab - b^2
   \]
   So, the combined fraction is:
   \[
   \frac{ab - b^2}{(a - b)(a + b)}
   \]

4. **Add the constant term:**

   Now add the constant \(3\) to the fraction:
   \[
   \frac{ab - b^2}{(a - b)(a + b)} + 3
   \]

   We can leave this as the final answer unless a further simplification is requested.

### Final Answer:
\[
\frac{ab - b^2}{(a - b)(a + b)} + 3
\]

Let me know if you'd like further details or clarifications!

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### Related Questions:
1. How can we simplify rational expressions more effectively?
2. What are common factoring techniques, like the difference of squares, and when do they apply?
3. How do we find a common denominator for more complex fractions?
4. How would this problem change if we were given numerical values for \(a\) and \(b\)?
5. How can we manipulate algebraic fractions involving polynomials?

### Tip:
When dealing with expressions that include a difference of squares, remember that \(a^2 - b^2\) always factors into \((a - b)(a + b)\), which can simplify many algebraic operations.

Perform the indicated operations for the expression: 2ab/(a^2 - b^2) - b/(a - b) + 6/2

Learn how to simplify a rational expression involving terms like 2ab/(a^2 - b^2), with a detailed step-by-step solution using factoring techniques such as the difference of squares. Suitable for students in Grades 10-12.

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