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To solve this problem using synthetic division, let's go through the steps carefully. We are dividing \( 2x^3 + 3x^2 - x - 4 \) by \( x + 1 \).

### Step 1: Set Up Synthetic Division
1. Write down the coefficients of the polynomial: \( 2, 3, -1, -4 \).
2. Since we are dividing by \( x + 1 \), we take the opposite of \( +1 \), which is \( -1 \), as our divisor.

### Step 2: Perform Synthetic Division
- **First row**: Write the coefficients \( 2, 3, -1, -4 \).
- **Second row**: This row will hold the results as we go through the division.

Here's the process:

1. **Bring down** the first coefficient, \( 2 \), to the second row.
2. **Multiply** \( -1 \) (the divisor) by \( 2 \), giving \( -2 \). Add this to the next coefficient, \( 3 \), yielding \( 1 \).
3. **Multiply** \( -1 \) by \( 1 \), giving \( -1 \). Add this to the next coefficient, \( -1 \), yielding \( -2 \).
4. **Multiply** \( -1 \) by \( -2 \), giving \( 2 \). Add this to the final coefficient, \( -4 \), yielding \( -2 \).

The result of the synthetic division is:
- Quotient coefficients: \( 2, 1, -2 \)
- Remainder: \( -2 \)

### Step 3: Write the Quotient
The quotient of \( 2x^3 + 3x^2 - x - 4 \) divided by \( x + 1 \) is:
\[
2x^2 + x - 2 + \frac{-2}{x + 1}
\]

So, the completed answer is:
\[
2x^2 + x - 2 + \frac{-2}{x + 1}
\]

### Final Answer:
- The values in the boxes are:
  - First box (coefficient of \( x^2 \)): \( 2 \)
  - Second box (coefficient of \( x \)): \( 1 \)
  - Third box (constant term): \( -2 \)
  - Fourth box (remainder): \( -2 \)

Would you like more details on synthetic division, or have any questions about this solution?

### Additional Questions:
1. How does synthetic division differ from long division of polynomials?
2. What are some common applications of synthetic division?
3. How would the answer change if the divisor were \( x - 1 \) instead of \( x + 1 \)?
4. Can synthetic division be used to factor polynomials?
5. How can we verify the result of synthetic division?

### Tip:
Synthetic division is a fast way to divide polynomials by linear terms, and it’s particularly useful for finding roots and simplifying expressions.

Use synthetic division to find the quotient: (2x^3 + 3x^2 - x - 4) ÷ (x + 1).

Learn how to use synthetic division to divide the polynomial 2x^3 + 3x^2 - x - 4 by x + 1. This step-by-step solution explains synthetic division and shows how to identify the quotient and remainder. Suitable for high school students studying algebra.

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