To find the domain of the function $f(x) = \frac{x^2 - 16}{x^2 + x - 6}$, we need to determine where the function is undefined, which occurs when the denominator equals zero.

Steps:

1. Denominator: $x^2 + x - 6 = 0$
2. Factor the quadratic expression: $x^2 + x - 6 = (x - 2)(x + 3) = 0$
3. Solve for $x$: $x - 2 = 0 \quad \Rightarrow \quad x = 2$ $x + 3 = 0 \quad \Rightarrow \quad x = -3$

Thus, the function is undefined at $x = 2$ and $x = -3$, because these values make the denominator zero.

Conclusion:

The domain of the function $f(x)$ is all real numbers except $x = 2$ and $x = -3$.

So, the domain is: $\boxed{(-\infty, -3) \cup (-3, 2) \cup (2, \infty)}$

Would you like further clarification or details on any step?

Here are 5 questions to deepen your understanding:

1. What happens to the function when the denominator is zero?
2. How do you factor a quadratic equation?
3. Can a function have more than two points where it's undefined?
4. How does the domain of a rational function compare to that of a polynomial function?
5. How would the domain change if the numerator were different?

Tip: Always check for factors common to both the numerator and denominator—they can affect the behavior of the function!