If (x+a) is a factor of `f(x)=x^3+ax^2-2x+a+4` , then a is

Options

A. `4/3`
B. `-2/3`
C. `-4/3`
D. `-7/3`

Answer

To find the value of \( a \) such that \( (x + a) \) is a factor of the polynomial \( f(x) = x^3 + ax^2 - 2x + a + 4 \), we can use the Factor Theorem. According to the Factor Theorem, if \( (x + a) \) is a factor of \( f(x) \), then \( f(-a) = 0 \).

### Step-by-Step Solution:

1. **Substitute \( x = -a \) into \( f(x) \)**:
\[ f(-a) = (-a)^3 + a(-a)^2 - 2(-a) + a + 4 \]

2. **Calculate each term**:
- \( (-a)^3 = -a^3 \)
- \( a(-a)^2 = a \cdot a^2 = a^3 \)
- \( -2(-a) = 2a \)

Thus, substituting these values back into the equation gives:
\[ f(-a) = -a^3 + a^3 + 2a + a + 4 \]

3. **Combine like terms**:
\[ f(-a) = (-a^3 + a^3) + (2a + a) + 4 = 0 + 3a + 4 \]

4. **Set the equation to zero**:
\[ 3a + 4 = 0 \]

5. **Solve for \( a \)**:
\[ 3a = -4 \quad \Rightarrow \quad a = -\frac{4}{3} \]

### Final Answer:
Thus, the value of \( a \) is:
\[ \boxed{-\frac{4}{3}} \]