To determine the domain of the function given:

$f(x) = \frac{x - x^2}{x^2 + 3x + 2}$

We need to identify the values of $x$ that make the denominator zero, as the function is undefined for those values.

Step 1: Factor the denominator

The denominator is $x^2 + 3x + 2$. We can factor it as follows:

$x^2 + 3x + 2 = (x + 1)(x + 2)$

Step 2: Set the denominator equal to zero

To find the values that make the denominator zero, solve the equation:

$(x + 1)(x + 2) = 0$

This gives us two solutions:

$x = -1 \quad \text{or} \quad x = -2$

Step 3: Domain of the function

The function is undefined at $x = -1$ and $x = -2$, so the domain of the function is all real numbers except $x = -1$ and $x = -2$.

Conclusion:

The domain of $f(x)$ is:

$\boxed{(-\infty, -2) \cup (-2, -1) \cup (-1, \infty)}$

Would you like more details on factoring, or how the domain was determined?

Here are some related questions:

1. How do you factor quadratic expressions like $x^2 + 3x + 2$?
2. What are the general steps to determine the domain of a rational function?
3. How do we handle undefined points in a function?
4. What happens if the numerator of the function equals zero at the undefined points?
5. How do you graph the function $f(x) = \frac{x - x^2}{x^2 + 3x + 2}$?

Tip: Always check for zeros in the denominator when determining the domain of a rational function!