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To determine the correct expression for the remainder when dividing \(2x^3 + 4x^2 + 5x - 1\) by \(x + 1\), we can apply the **Remainder Theorem**.

### Solution:

According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x - a\), the remainder of this division is \(f(a)\).

In this case:
- \(f(x) = 2x^3 + 4x^2 + 5x - 1\)
- The divisor is \(x + 1\), which we can rewrite as \(x - (-1)\), so \(a = -1\).

1. **Evaluate \(f(-1)\):**
   \[
   f(-1) = 2(-1)^3 + 4(-1)^2 + 5(-1) - 1
   \]
2. **Simplify each term:**
   - \(2(-1)^3 = 2(-1) = -2\)
   - \(4(-1)^2 = 4(1) = 4\)
   - \(5(-1) = -5\)
   - \(-1\)

3. **Combine the results:**
   \[
   f(-1) = -2 + 4 - 5 - 1 = -4
   \]

### Conclusion:
The remainder when dividing \(2x^3 + 4x^2 + 5x - 1\) by \(x + 1\) is indeed \(-4\). 

Thus, the correct answer is:
- **Option 1: \(-4\)**

Would you like more details, or have any questions about the solution?

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### Related Questions:
1. How would the remainder change if the divisor were \(x - 2\) instead?
2. Can we use synthetic division to confirm the remainder for this problem?
3. How does the Remainder Theorem apply when the divisor is a quadratic expression?
4. What are the steps to perform polynomial long division for this problem?
5. How does the Remainder Theorem help in evaluating polynomial functions quickly?

### Tip:
The Remainder Theorem is a quick way to find the remainder without performing full polynomial division, especially when the divisor is in the form \(x - a\).

The polynomial long division of 2x^3 + 4x^2 + 5x - 1 by x + 1 results in a remainder of -4. Which correctly expresses this remainder?

Learn how to find the remainder when dividing the polynomial 2x^3 + 4x^2 + 5x - 1 by x + 1. This example illustrates the use of the Remainder Theorem and polynomial division concepts, suitable for high school students in Grades 9-12.

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