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To solve the problem, let's perform polynomial division on the given expression:

\[
\frac{6x^3 - 3x^2 - 13x + 6}{2x - 3}
\]

We'll use long division to find the quotient and remainder.

### Step 1: Divide the leading terms
- Divide \(6x^3\) by \(2x\), which gives \(3x^2\).
- Multiply \(3x^2\) by \(2x - 3\), resulting in \(6x^3 - 9x^2\).
- Subtract \((6x^3 - 9x^2)\) from the original polynomial.

This yields:
\[
6x^3 - 3x^2 - 13x + 6 - (6x^3 - 9x^2) = 6x^2 - 13x + 6
\]

### Step 2: Repeat the process for the next term
- Divide \(6x^2\) by \(2x\), which gives \(3x\).
- Multiply \(3x\) by \(2x - 3\), resulting in \(6x^2 - 9x\).
- Subtract \((6x^2 - 9x)\) from the result.

This yields:
\[
6x^2 - 13x + 6 - (6x^2 - 9x) = -4x + 6
\]

### Step 3: Repeat the process for the last term
- Divide \(-4x\) by \(2x\), which gives \(-2\).
- Multiply \(-2\) by \(2x - 3\), resulting in \(-4x + 6\).
- Subtract \((-4x + 6)\) from the result.

This yields:
\[
-4x + 6 - (-4x + 6) = 0
\]

Since there is no remainder, the division is complete, and the answer is:

\[
3x^2 + 3x - 2
\]

Let me know if you'd like a detailed explanation of any step or have further questions!

### Related Questions
1. How do you perform synthetic division on a polynomial?
2. What happens if there is a remainder after polynomial division?
3. How does polynomial division differ from regular division?
4. How can polynomial division be used in real-world applications?
5. What are other methods for simplifying polynomial expressions?

#### Tip
When dividing polynomials, always align terms by degree and subtract carefully to avoid mistakes.

Divide the polynomial \( \frac{6x^3 - 3x^2 - 13x + 6}{2x - 3} \) and give the answer in the form \( \text{quotient} + \frac{\text{remainder}}{\text{divisor}} \) if necessary.

Learn how to perform polynomial division for the expression \( \frac{6x^3 - 3x^2 - 13x + 6}{2x - 3} \) with step-by-step instructions, including the quotient and remainder if necessary. Suitable for high school students learning algebra and polynomial division techniques.

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