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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!

Déterminer le domaine de la fonction suivante : f(x) = (x - x^2) / (x^2 + 3x + 2)

Learn how to determine the domain of the rational function f(x) = (x - x^2) / (x^2 + 3x + 2). This step-by-step solution covers factoring the quadratic denominator and identifying values that make the denominator zero. Suitable for Grades 10-12.

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