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To solve the division of polynomials:

\[
\frac{-6x^2 - 3x + 12}{3x - 3}
\]

we will perform polynomial long division.

### Step-by-step solution:

1. **Divide the leading term** of the dividend by the leading term of the divisor:
   \[
   \frac{-6x^2}{3x} = -2x
   \]
   So, the first term of the quotient is \(-2x\).

2. **Multiply** the entire divisor \(3x - 3\) by \(-2x\):
   \[
   (-2x)(3x - 3) = -6x^2 + 6x
   \]
   
3. **Subtract** this from the original polynomial:
   \[
   (-6x^2 - 3x + 12) - (-6x^2 + 6x) = -9x + 12
   \]

4. **Repeat the process**: Divide the leading term of the new polynomial \(-9x\) by the leading term of the divisor \(3x\):
   \[
   \frac{-9x}{3x} = -3
   \]
   So, the next term of the quotient is \(-3\).

5. **Multiply** the entire divisor \(3x - 3\) by \(-3\):
   \[
   (-3)(3x - 3) = -9x + 9
   \]
   
6. **Subtract** again:
   \[
   (-9x + 12) - (-9x + 9) = 3
   \]

### Final result:
The quotient is:
\[
-2x - 3
\]
with a remainder of \(3\), so the division can be written as:
\[
-2x - 3 + \frac{3}{3x - 3}
\]
Simplified, this becomes:
\[
-2x - 3 + \frac{1}{x - 1}
\]

Would you like more details or further explanations on any of the steps?

Here are some related questions:
1. How do you perform synthetic division for this type of polynomial division?
2. What happens if the divisor is a more complex polynomial?
3. How can polynomial division be used to simplify rational expressions?
4. What is the relationship between polynomial division and factoring?
5. How do remainders in polynomial division relate to the Remainder Theorem?

**Tip:** Polynomial long division follows a very similar process to numerical long division, so understanding one can help with the other!

Learn how to divide the polynomial -6x^2 - 3x + 12 by 3x - 3 using polynomial long division. This step-by-step guide covers key algebraic concepts, including dividing terms, simplifying expressions, and handling remainders. Suitable for students in Grades 10-12.

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