To simplify the expression:

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

Step 1: Factor $x^2 - 1$

Notice that $x^2 - 1$ is a difference of squares:

$x^2 - 1 = (x - 1)(x + 1)$

Thus, the expression becomes:

$\frac{2}{(x - 1)(x + 1)} - \frac{1}{x - 1} + \frac{2}{x + 1}$

Step 2: Combine into a single fraction

To combine the fractions, find a common denominator. The common denominator is $(x - 1)(x + 1)$. Rewrite the fractions with this denominator:

$\frac{2}{(x - 1)(x + 1)} - \frac{1(x + 1)}{(x - 1)(x + 1)} + \frac{2(x - 1)}{(x - 1)(x + 1)}$

Step 3: Simplify the numerators

Now, simplify each numerator:

$\frac{2}{(x - 1)(x + 1)} - \frac{x + 1}{(x - 1)(x + 1)} + \frac{2(x - 1)}{(x - 1)(x + 1)}$

Simplifying the last term:

$\frac{2(x - 1)}{(x - 1)(x + 1)} = \frac{2x - 2}{(x - 1)(x + 1)}$

So, the expression becomes:

$\frac{2 - (x + 1) + (2x - 2)}{(x - 1)(x + 1)}$

Step 4: Combine like terms

Simplify the numerator:

$2 - (x + 1) + 2x - 2 = 2 - x - 1 + 2x - 2 = x - 1$

Thus, the final simplified expression is:

$\frac{x - 1}{(x - 1)(x + 1)}$

Step 5: Cancel common factors

Since $x - 1$ appears in both the numerator and denominator, we can cancel it out:

$\frac{1}{x + 1}$

Thus, the simplified form of the expression is:

$\frac{1}{x + 1}$

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Do you have any questions or need further clarification?

Here are five related questions:

1. How can you find the least common denominator (LCD) of rational expressions?
2. What is the process of simplifying complex rational expressions?
3. How do you factor differences of squares?
4. What are the rules for canceling common factors in rational expressions?
5. How does simplification help in solving rational equations?

Tip: Always check for common factors that can be canceled after factoring to simplify expressions more efficiently.