We are given the expression:

\frac{\frac{x^2 - 4}{x^2 + x - 2} \cdot \frac{x^2 - 1}{x^2 - x - 2}}

Step 1: Factorize the expressions

First, factor each quadratic term.

1. $x^2 - 4$ is a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$

2. $x^2 + x - 2$ factors as: $x^2 + x - 2 = (x - 1)(x + 2)$

3. $x^2 - 1$ is also a difference of squares: $x^2 - 1 = (x - 1)(x + 1)$

4. $x^2 - x - 2$ factors as: $x^2 - x - 2 = (x - 2)(x + 1)$

Step 2: Substitute factored forms into the expression

Now substitute the factored forms into the original expression:

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

Step 3: Simplify the expression

Cancel out the common terms from the numerator and denominator:

• $x - 2$ cancels.
• $x + 2$ cancels.
• $x - 1$ cancels.
• $x + 1$ cancels.

After all the cancellations, we are left with:

$1$

Thus, the simplified result is $1$.

Step 4: Choose the correct option

The answer corresponds to option b), which is $1$.

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How do you factor a quadratic expression?
2. What is the difference of squares formula?
3. How do you multiply rational expressions?
4. What are the steps to simplify rational expressions?
5. How do you determine restrictions on the variable in a rational expression?

Tip: Always check for restrictions (undefined values) in the variable when simplifying rational expressions by looking at the denominators.