Simplify 3x2−2x−1x2−3x+2÷3x2+7x+23x2+5x−2
.

To simplify the expression

\[
\frac{3x^2 - 2x - 1}{x^2 - 3x + 2} \div \frac{3x^2 + 7x + 2}{3x^2 + 5x - 2},
\]

we can follow these steps:

### Step 1: Express the division as multiplication by the reciprocal
The given expression can be rewritten as:

\[
\frac{3x^2 - 2x - 1}{x^2 - 3x + 2} \times \frac{3x^2 + 5x - 2}{3x^2 + 7x + 2}
\]

### Step 2: Factor each quadratic expression (if possible)
Let's factor the quadratics:

1. \( 3x^2 - 2x - 1 \) factors into \( (3x + 1)(x - 1) \).
2. \( x^2 - 3x + 2 \) factors into \( (x - 1)(x - 2) \).
3. \( 3x^2 + 7x + 2 \) factors into \( (3x + 1)(x + 2) \).
4. \( 3x^2 + 5x - 2 \) factors into \( (3x - 1)(x + 2) \).

Thus, the expression becomes:

\[
\frac{(3x + 1)(x - 1)}{(x - 1)(x - 2)} \times \frac{(3x - 1)(x + 2)}{(3x + 1)(x + 2)}
\]

### Step 3: Simplify the expression by canceling out common factors
We can cancel the common factors in the numerator and denominator:

- \( (3x + 1) \) cancels with \( (3x + 1) \).
- \( (x - 1) \) cancels with \( (x - 1) \).
- \( (x + 2) \) cancels with \( (x + 2) \).

This leaves us with:

\[
\frac{3x - 1}{x - 2}
\]

### Final Answer
The simplified expression is:

\[
\frac{3x - 1}{x - 2}
\]

Would you like to go over any step in detail or have any questions?

Here are some additional related problems:
1. Simplify \(\frac{x^2 - 4x + 4}{x^2 - 4}\).
2. Factor \(2x^2 + 7x + 3\).
3. Simplify \(\frac{x^2 - 9}{x^2 - x - 6}\).
4. Solve the equation \(\frac{3x - 1}{x - 2} = 0\).
5. Factor \(x^2 - 5x + 6\).
6. Simplify \(\frac{4x^2 - 1}{2x - 1}\).
7. Solve \(\frac{3x - 1}{x - 2} = 2\).
8. Factor \(4x^2 - 12x + 9\).

**Tip:** Always check if quadratic expressions can be factored to simplify rational expressions more easily.

Learn how to simplify the complex fraction 3x^2 - 2x - 1 / x^2 - 3x + 2 divided by 3x^2 + 7x + 2 / 3x^2 + 5x - 2 using step-by-step methods. This problem involves algebraic concepts and fraction simplification. Suitable for students in Grades 9-12.

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