https://file.aiquickdraw.com/mathbot/file/1729829328800bd1l3xqa.jpg

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."

Simplify the expression \(\frac{x^2 - 4}{x^2 - 2x - 8}\).

Step-by-step guide to simplify the rational expression (x^2 - 4) / (x^2 - 2x - 8). Learn how to factor the numerator and denominator and cancel common factors. Suitable for students in Grades 8-10.

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