https://file.aiquickdraw.com/mathbot/file/1728324664599dj49cmr1.jpg

The question asks to divide the polynomial \(2x^3 - 3x^2 - 17x + 30\) by \(x + 3\).

We can use polynomial long division or synthetic division to solve this. Let's proceed with synthetic division since \(x + 3\) has the form \(x - r\) where \(r = -3\).

### Step-by-Step Synthetic Division:

1. **Coefficients of the dividend:**  
   The polynomial \(2x^3 - 3x^2 - 17x + 30\) has coefficients:  
   \[ 2, -3, -17, 30 \]
   
2. **Synthetic division setup:**  
   The divisor is \(x + 3\), so we use \(r = -3\) for synthetic division.  
   
   ```
   -3 |  2    -3    -17    30
       |        -6    27    -30
       ---------------------
         2    -9    10      0
   ```

   - Bring down the 2.
   - Multiply \(2 \times -3 = -6\), add to the next coefficient \(-3 + (-6) = -9\).
   - Multiply \(-9 \times -3 = 27\), add to the next coefficient \(-17 + 27 = 10\).
   - Multiply \(10 \times -3 = -30\), add to the last coefficient \(30 + (-30) = 0\).

3. **Quotient:**  
   The quotient is \(2x^2 - 9x + 10\) with a remainder of 0, meaning the division is exact.

### Answer:  
The result of dividing \(2x^3 - 3x^2 - 17x + 30\) by \(x + 3\) is \(2x^2 - 9x + 10\).

The correct answer is:  
**\(2x^2 - 9x + 10\)** (which is marked in the image).

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Do you need further explanation or have any questions?

Here are 5 related questions for further practice:
1. How would you divide \(3x^3 + 5x^2 - 2x - 8\) by \(x + 4\)?
2. What is synthetic division, and when is it useful?
3. Can you divide \(4x^4 + 6x^3 - 9x + 15\) by \(x - 1\) using synthetic division?
4. What is the relationship between the quotient and remainder in polynomial division?
5. How do you know if a polynomial is divisible by another polynomial without a remainder?

**Tip:** When using synthetic division, always ensure the polynomial is in standard form (i.e., terms in descending order of degree) and check for missing terms.

What is the result of dividing 2x^3 - 3x^2 - 17x + 30 by x + 3?

This page explains how to divide the polynomial 2x^3 - 3x^2 - 17x + 30 by x + 3 using synthetic division. Learn the step-by-step process, including setting up the synthetic division and solving for the quotient and remainder. Suitable for high school students in Grades 9-12.

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