If x+1 is a factor of `ax^(3)+x^(2)-2x+4a-9`, then value of a .

Options

A. -9
B. 0
C. 6
D. 2

Explanation

To find the value of \( a \) such that \( x + 1 \) is a factor of the polynomial \( ax^3 + x^2 - 2x + 4a - 9 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Polynomial**:
We have the polynomial \( P(x) = ax^3 + x^2 - 2x + 4a - 9 \).

2. **Use the Factor Theorem**:
Since \( x + 1 \) is a factor, it implies that \( P(-1) = 0 \).

3. **Substitute \( x = -1 \)**:
We substitute \( -1 \) into the polynomial:
\[ P(-1) = a(-1)^3 + (-1)^2 - 2(-1) + 4a - 9 \]

4. **Simplify the Expression**:
Calculate each term:
\[ P(-1) = a(-1) + 1 + 2 + 4a - 9 \]
This simplifies to:
\[ P(-1) = -a + 1 + 2 + 4a - 9 \]
Combine like terms:
\[ P(-1) = (-a + 4a) + (1 + 2 - 9) = 3a - 6 \]

5. **Set the Polynomial Equal to Zero**:
Since \( P(-1) = 0 \):
\[ 3a - 6 = 0 \]

6. **Solve for \( a \)**:
Rearranging gives:
\[ 3a = 6 \]
Dividing both sides by 3:
\[ a = 2 \]

### Conclusion:
The value of \( a \) is \( 2 \).