When `2f^(3)+3f^(2)-1` is divided by `f+2`, the remainder is

Options

A. `-9`
B. `-5`
C. `-1`
D. 2

Explanation

To find the remainder when \( 2f^3 + 3f^2 - 1 \) is divided by \( f + 2 \), we can use the Remainder Theorem. According to the theorem, the remainder of a polynomial \( P(f) \) when divided by \( f - c \) is \( P(c) \). In our case, we need to evaluate \( P(-2) \) since we are dividing by \( f + 2 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- Polynomial: \( P(f) = 2f^3 + 3f^2 - 1 \)
- Divisor: \( f + 2 \) (which means we will evaluate at \( f = -2 \))

2. **Substitute \( f = -2 \) into the polynomial**:
\[ P(-2) = 2(-2)^3 + 3(-2)^2 - 1 \]

3. **Calculate \( (-2)^3 \) and \( (-2)^2 \)**:
- \( (-2)^3 = -8 \)
- \( (-2)^2 = 4 \)

4. **Substitute these values back into the polynomial**:
\[ P(-2) = 2(-8) + 3(4) - 1 \]

5. **Perform the multiplication**:
\[ P(-2) = -16 + 12 - 1 \]

6. **Combine the terms**:
\[ P(-2) = -16 + 12 = -4 \]
\[ P(-2) = -4 - 1 = -5 \]

7. **Final result**:
The remainder when \( 2f^3 + 3f^2 - 1 \) is divided by \( f + 2 \) is \( -5 \).

### Summary:
The remainder is \( -5 \).