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This problem involves polynomial division. We need to divide \(-8x^3 + 12x^2 + 14x - 21\) by \(-2x + 3\) using long division, and then fill in the missing boxes in the given diagram.

### Steps to solve this:

1. **Divide the first terms**:
   Divide \(-8x^3\) by \(-2x\). This gives \(4x^2\), which we place in the top-left box.

2. **Multiply**:
   Multiply \(4x^2\) by the divisor \(-2x + 3\). This gives:
   \[
   4x^2 \times (-2x) = -8x^3 \quad \text{(this goes under the \(-8x^3\))} 
   \]
   \[
   4x^2 \times 3 = 12x^2 \quad \text{(this goes under the \(12x^2\))}
   \]
   These values fill the first row of the multiplication in the table.

3. **Subtract**:
   Subtract \((-8x^3 + 12x^2)\) from \((-8x^3 + 12x^2 + 14x)\), leaving:
   \[
   0x^3 + 0x^2 + 14x \quad \text{(14x remains, so move that down)}
   \]

4. **Divide the next terms**:
   Divide \(14x\) by \(-2x\). This gives \(-7\), which we place in the top-right box.

5. **Multiply**:
   Multiply \(-7\) by the divisor \(-2x + 3\):
   \[
   -7 \times (-2x) = 14x \quad \text{(this goes under the \(14x\))}
   \]
   \[
   -7 \times 3 = -21 \quad \text{(this goes under the \(-21\))}
   \]
   This completes the multiplication and the final row of subtraction, leaving \(0\).

### Final Values:
- The box to the right of \(4x^2\) is \(-2x\),
- The box under \(-2x\) is \(-8x^3\),
- The box under \(14x\) is \(14x\),
- The box next to \(-7\) is \(-21\),
- The box under \(12x^2\) is \(0\).

The quotient is \(4x^2 - 2x - 7\).

Would you like more details on any step or have any questions?

Here are 5 related questions to extend this:
1. How would the solution change if the divisor was \(2x + 3\)?
2. Can synthetic division be used here, and how?
3. How do you handle division when the leading coefficients don’t cancel directly?
4. What are other methods to verify the quotient besides long division?
5. How does the remainder theorem relate to this type of division?

**Tip**: Always check your work by multiplying the quotient and divisor to ensure you get back the original polynomial.

Fill in the missing values below one at a time to find the quotient when -8x^3 + 12x^2 + 14x - 21 is divided by -2x + 3.

Learn how to perform polynomial long division to divide -8x^3 + 12x^2 + 14x - 21 by -2x + 3. This step-by-step guide explains key algebraic techniques and is suitable for students in Grades 9-12.

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