The expression you've provided is:

$\frac{x^2 - 4}{x^2 - 2x - 8}$

Let's break it down and simplify step by step:

Step 1: Factor both the numerator and denominator

1. Numerator: $x^2 - 4$ is a difference of squares, so it can be factored as: $x^2 - 4 = (x - 2)(x + 2)$

2. Denominator: $x^2 - 2x - 8$ can be factored by finding two numbers that multiply to $-8$ and add to $-2$. These numbers are $-4$ and $2$. So, we factor the denominator as: $x^2 - 2x - 8 = (x - 4)(x + 2)$

Step 2: Simplify the expression

Now we substitute the factored forms into the original expression: $\frac{(x - 2)(x + 2)}{(x - 4)(x + 2)}$

We can cancel the common factor $(x + 2)$ from the numerator and denominator (assuming $x \neq -2$, to avoid division by zero).

Thus, the simplified expression is: $\frac{x - 2}{x - 4}$

Final Answer:

$\frac{x - 2}{x - 4}, \quad x \neq -2$

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How do you factor the difference of squares in general?
2. What happens if $x = -2$ in the original expression?
3. Can you always cancel terms in a rational expression?
4. How do you determine which values make a denominator undefined?
5. What are other methods to simplify rational expressions?

Tip: Always check for values that make the denominator zero when simplifying rational expressions. These are called "excluded values."