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Let's solve this division problem using synthetic division.

The expression to divide is \( (2x^2 - 3x - 9) \div (x - 3) \).

### Steps for Synthetic Division:

1. **Identify coefficients** from the dividend \( 2x^2 - 3x - 9 \). The coefficients are \( [2, -3, -9] \).

2. **Change the sign** of the root of the divisor \( x - 3 \). Here, the root is \( 3 \), so we'll use \( 3 \) for the synthetic division.

3. **Set up the synthetic division:**

```
  3 |   2   -3   -9
      |       6    9
      -----------------
         2    3    0
```

### Explanation:
- Bring down the first coefficient (2) unchanged.
- Multiply \( 2 \times 3 = 6 \), and place it under the next coefficient.
- Add \( -3 + 6 = 3 \).
- Multiply \( 3 \times 3 = 9 \), and place it under the next coefficient.
- Add \( -9 + 9 = 0 \).

### Final quotient:

The quotient is \( 2x + 3 \), with a remainder of 0.

### Conclusion:

The correct answer is **d) 2x + 3**.

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Would you like more details or have any questions?

Here are 5 related questions you might want to explore:

1. What is the synthetic division process when dividing by a quadratic divisor?
2. How does synthetic division compare with long division of polynomials?
3. What happens when there’s a remainder in synthetic division?
4. Can synthetic division be used when the divisor's leading coefficient is not 1?
5. How does synthetic division apply when dealing with higher degree polynomials?

**Tip**: Synthetic division only works when the divisor is a linear polynomial of the form \( x - r \).

Learn how to solve the synthetic division problem (2x^2 - 3x - 9) ÷ (x - 3). This problem covers key algebraic concepts, including synthetic division and polynomial division, suitable for high school students in Grades 9-12.

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