To simplify the expression:

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

Step 1: Recognize a common denominator

The denominators in both fractions are the same: $a^2 - b^2$. We can factor this as a difference of squares:

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

Thus, the expression becomes:

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

Step 2: Combine the numerators

Since the denominators are the same, we can combine the two fractions:

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

Step 3: Simplify the numerator

Simplify the numerator by distributing the negative sign in the second term:

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

Step 4: Final simplified expression

Now substitute the simplified numerator:

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

Thus, the simplified expression is:

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

Would you like further details or any clarifications?

Here are five related questions:

1. How do you factor more complex quadratic expressions?
2. What is the difference of squares identity used for?
3. Can this expression be simplified further if $b = 0$?
4. What happens when $a = b$?
5. How do you solve an equation involving fractions with polynomial denominators?

Tip: Always check for common factors in numerators and denominators to simplify fractions.