To find the points of discontinuity for the function:

$f(x) = \frac{x e^x}{x^2 + x - 2}$

we need to identify values of $x$ where the denominator equals zero, as these values make the function undefined.

Step 1: Factor the Denominator

The denominator is: $x^2 + x - 2$

To factor this expression, we look for two numbers that multiply to $-2$ and add up to $1$. These numbers are $2$ and $-1$, so we can rewrite the denominator as: $x^2 + x - 2 = (x + 2)(x - 1)$

Step 2: Set the Denominator Equal to Zero

Now, we find the values of $x$ that make the denominator zero: $(x + 2)(x - 1) = 0$

This gives: $x = -2 \quad \text{and} \quad x = 1$

Conclusion

The function $f(x) = \frac{x e^x}{x^2 + x - 2}$ is undefined at $x = -2$ and $x = 1$. Therefore, these are the points of discontinuity.

Would you like to see further details on the type of discontinuities (such as removable or infinite discontinuities) or have any other questions?

Here are some related questions for further exploration:

1. How do we determine if a discontinuity is removable or non-removable?
2. What happens to $f(x)$ as $x$ approaches the points of discontinuity?
3. How does the factorization of the denominator help in analyzing rational functions?
4. What other types of functions have discontinuities besides rational functions?
5. How would the discontinuity change if we added a constant term to the function?

Tip: When analyzing a rational function, always start by factoring the denominator to identify potential discontinuities.