If `omega` is complex cube root of unity ans `x=omega^(2)-omega-2,` then what is the vlaue of `x^(2)+4x+7` ?

Options

A. `-2`
B. `-1`
C. 0
D. 1

Explanation

To solve the problem, we need to find the value of \( x^2 + 4x + 7 \) given that \( x = \omega^2 - \omega - 2 \), where \( \omega \) is a complex cube root of unity. The cube roots of unity satisfy the equation \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

### Step-by-Step Solution:

1. **Identify the value of \( x \)**:
\[ x = \omega^2 - \omega - 2 \]

2. **Rearranging \( x \)**:
\[ x + 2 = \omega^2 - \omega \]

3. **Square both sides**:
\[ (x + 2)^2 = (\omega^2 - \omega)^2 \]

4. **Expand both sides**:
\[ x^2 + 4x + 4 = \omega^4 - 2\omega^3 + \omega^2 \]

5. **Substituting values of powers of \( \omega \)**:
- Since \( \omega^3 = 1 \), we have \( \omega^4 = \omega \).
- Thus, substituting gives:
\[ x^2 + 4x + 4 = \omega - 2 \cdot 1 + \omega^2 \]
\[ x^2 + 4x + 4 = \omega + \omega^2 - 2 \]

6. **Using the property of cube roots of unity**:
- From \( 1 + \omega + \omega^2 = 0 \), we can express \( \omega + \omega^2 \) as:
\[ \omega + \omega^2 = -1 \]
- Therefore, substituting this into our equation gives:
\[ x^2 + 4x + 4 = -1 - 2 = -3 \]

7. **Finding \( x^2 + 4x + 7 \)**:
\[ x^2 + 4x + 7 = (x^2 + 4x + 4) + 3 = -3 + 3 = 0 \]

### Final Answer:
\[ x^2 + 4x + 7 = 0 \]