The function you've provided is:

\[
f(x) = \frac{x^2 + x - 6}{x^2 - 4}
\]

To analyze this function, we can do the following:

### Step 1: Factor the numerator and the denominator
Let's factor both the numerator and denominator to simplify the function.

#### Numerator: \( x^2 + x - 6 \)
We need two numbers that multiply to \(-6\) and add up to \(1\) (the coefficient of \(x\)). The numbers are \(3\) and \(-2\), so:

\[
x^2 + x - 6 = (x + 3)(x - 2)
\]

#### Denominator: \( x^2 - 4 \)
This is a difference of squares, which factors as:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

### Step 2: Simplify the function
Now, the function becomes:

\[
f(x) = \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)}
\]

We can cancel out the common factor \( (x - 2) \) in the numerator and denominator, but we must note that \( x \neq 2 \), as this would make the denominator zero and cause the function to be undefined at \( x = 2 \).

The simplified function is:

\[
f(x) = \frac{x + 3}{x + 2}, \quad x \neq 2
\]

### Step 3: Identify any restrictions
- The function is undefined at \( x = 2 \) and \( x = -2 \) because the original denominator becomes zero at those points.
- There is a **vertical asymptote** at \( x = -2 \), where the denominator equals zero but no cancellation occurs.
- At \( x = 2 \), there is a **hole** in the graph because the factor \( x - 2 \) was canceled out.

### Summary:
The simplified function is:

\[
f(x) = \frac{x + 3}{x + 2}, \quad x \neq 2, x \neq -2
\]

- Vertical asymptote at \( x = -2 \)
- Hole at \( x = 2 \)

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Would you like more details on how these concepts (asymptotes, holes) are derived?

### Related Questions:
1. How do you find horizontal asymptotes of rational functions?
2. How do you graph rational functions with vertical asymptotes and holes?
3. What happens to the function near \( x = -2 \) and \( x = 2 \)?
4. How does the function behave as \( x \to \infty \)?
5. Can rational functions have more than one vertical asymptote?

### Tip:
When simplifying rational functions, always check for points where the original function is undefined, even after canceling terms.

Learn how to simplify the rational function f(x) = (x^2 + x - 6) / (x^2 - 4). This solution includes factoring, identifying vertical asymptotes, and determining holes in the graph. Suitable for students in Grades 9-12.

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