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To solve this problem using synthetic division, we need to divide \( 2x^4 - 3x^3 - 4x + 7 \) by \( x + 3 \).

### Steps for Synthetic Division:
1. **Set up the coefficients** of the polynomial: 
   \[
   2, -3, 0, -4, 7
   \]
   Notice that there's no \( x^2 \) term, so its coefficient is 0.

2. **Change the sign of the divisor.** Since we're dividing by \( x + 3 \), we'll use \( -3 \).

3. **Perform the synthetic division** using \( -3 \) and the coefficients.

### Synthetic Division Calculation:

1. **Bring down the leading coefficient** \( 2 \).

2. Multiply \( 2 \) by \( -3 \) to get \( -6 \). Place \( -6 \) under the next coefficient.

3. Add \( -3 + (-6) = -9 \).

4. Multiply \( -9 \) by \( -3 \) to get \( 27 \). Place \( 27 \) under the next coefficient (0).

5. Add \( 0 + 27 = 27 \).

6. Multiply \( 27 \) by \( -3 \) to get \( -81 \). Place \( -81 \) under the next coefficient (-4).

7. Add \( -4 + (-81) = -85 \).

8. Multiply \( -85 \) by \( -3 \) to get \( 255 \). Place \( 255 \) under the next coefficient (7).

9. Add \( 7 + 255 = 262 \).

### Result:

The resulting coefficients are:
\[
2, -9, 27, -85, 262
\]

This corresponds to the final row of the synthetic division setup. Checking against the given options:

- The correct answer is the last option, which shows the coefficients of the result as \( 2, -9, 27, -85, 262 \).

Let me know if you need more details or have any questions. Here are five related questions to consider:

1. What is the remainder when dividing \( 2x^4 - 3x^3 - 4x + 7 \) by \( x + 3 \)?
2. How do you interpret the result of synthetic division?
3. How would you solve a similar problem using long division instead of synthetic division?
4. Can synthetic division be used for all polynomial divisions?
5. What is the relationship between the roots of the divisor and the quotient obtained?

**Tip:** Always remember to include missing coefficients as zeros in synthetic division to ensure accurate calculations.

Which of the following shows the correct illustration when 2x^4 - 3x^3 - 4x + 7 is divided by x + 3 using synthetic division?

Learn how to perform synthetic division for the polynomial 2x^4 - 3x^3 - 4x + 7 divided by x + 3. This detailed step-by-step solution covers algebraic concepts and polynomial division. Suitable for students in Grades 10-12.

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