The function given is:

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

To simplify and analyze this function, let's break it down:

### 1. Factorization:
First, let's factor both the numerator and the denominator if possible.

- **Numerator:** \( 3x^3 + 2x^2 \) can be factored as \( x^2(3x + 2) \).
- **Denominator:** \( x^2 - x \) can be factored as \( x(x - 1) \).

So, the function can be rewritten as:

\[
f(x) = \frac{x^2(3x + 2)}{x(x - 1)}
\]

### 2. Simplification:
We can simplify the function by canceling out a common factor of \( x \) from the numerator and the denominator, but note that \( x \) should not be 0, as it would make the denominator zero.

\[
f(x) = \frac{x(3x + 2)}{x - 1} \quad \text{for} \quad x \neq 0
\]

### 3. Domain:
The function \( f(x) \) is undefined where the denominator is zero. So, \( x - 1 = 0 \), which gives \( x = 1 \). Additionally, \( x = 0 \) is also excluded because we canceled \( x \) during simplification.

Thus, the domain of \( f(x) \) is all real numbers except \( x = 0 \) and \( x = 1 \).

### 4. Final Simplified Function:
After simplification, the function is:

\[
f(x) = \frac{3x^2 + 2x}{x - 1}
\]

for \( x \neq 0 \) and \( x \neq 1 \).

Would you like more details or have any questions?

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**Related Questions:**

1. How do you find the limits of \( f(x) \) as \( x \) approaches 0 and 1?
2. What are the vertical asymptotes of the function \( f(x) \)?
3. How would you differentiate the simplified function \( f(x) = \frac{3x^2 + 2x}{x - 1} \)?
4. What is the behavior of \( f(x) \) as \( x \) approaches infinity?
5. How do you sketch the graph of the simplified function \( f(x) \)?

**Tip:** Always check for factors common to the numerator and denominator before simplifying a rational function.

Learn how to simplify and analyze the rational function (3x^3 + 2x^2) / (x^2 - x) step-by-step. This problem covers rational function simplification, factorization techniques, and domain analysis. Suitable for advanced high school students and college-level learners.

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