Let's analyze the function \( h(x) = \frac{x^2 - 2x - 8}{x - 2} \).

### Step 1: Factor the Numerator
The numerator is a quadratic expression, \( x^2 - 2x - 8 \). Let's factor it:
\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]
So the function can be rewritten as:
\[
h(x) = \frac{(x - 4)(x + 2)}{x - 2}
\]

### Step 2: Simplify the Function
The expression can be simplified by canceling out common factors. However, we must consider the domain of the function to ensure no division by zero occurs.

The denominator \( x - 2 \) implies that \( x \) cannot be 2 because the function would be undefined at that point.

Thus, the simplified function is:
\[
h(x) = x + 2 \quad \text{for } x \neq 2
\]
However, the function is **undefined** at \( x = 2 \).

### Conclusion:
- The simplified form of the function is \( h(x) = x + 2 \), with the condition that \( x \neq 2 \).
- There is a **hole** in the graph of the function at \( x = 2 \) because the function is undefined at this point.

Would you like to discuss any particular aspect of this function or need more details?

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Here are some related questions you might consider:

1. What is the domain of \( h(x) \)?
2. How does the graph of \( h(x) \) look, and where is the hole located?
3. What happens to \( h(x) \) as \( x \) approaches 2?
4. Can we find the limit of \( h(x) \) as \( x \) approaches 2?
5. How would the function change if the denominator were different?

**Tip:** When simplifying rational functions, always check for points where the denominator is zero to determine the domain and identify any holes or vertical asymptotes.

Explore the rational function h(x) = (x^2 - 2x - 8) / (x - 2) step-by-step. Learn how to factor the numerator, simplify the function, and understand its domain limitations. This problem involves algebraic manipulation and rational function analysis, suitable for Grades 10-12 students.

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