The expression you've provided is:

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

Step 1: Factor both the numerator and denominator.

• Numerator $x^2 - x - 2$ can be factored as: $x^2 - x - 2 = (x - 2)(x + 1)$

• Denominator $x^2 - 1$ is a difference of squares: $x^2 - 1 = (x - 1)(x + 1)$

Step 2: Rewrite the expression using the factored forms.

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

Step 3: Simplify the expression by canceling out common factors.

The factor $(x + 1)$ appears in both the numerator and denominator, so we cancel it out:

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

Final Answer:

The simplified form of the given expression is:

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

Do you have any further questions or would you like additional details?

Here are 5 related questions you might explore:

1. How do you factor a quadratic expression?
2. What is the difference of squares and how is it factored?
3. Can this expression be simplified further if certain values of $x$ are given?
4. Are there any restrictions on $x$ in this simplified expression?
5. How does factoring help in solving rational equations?

Tip: Always check for common factors when simplifying rational expressions to avoid unnecessary complexity.