If `x=omega-omega^2-2` then , the value of `x^4+3x^3+2x^2-11x-6` is (where `omega ` is a imaginary cube root of unity)

Explanation

To solve the problem step by step, we start with the given expression and the properties of the imaginary cube roots of unity.

### Step 1: Understand the properties of cube roots of unity
The cube roots of unity are given by:
- \( \omega^3 = 1 \)
- \( 1 + \omega + \omega^2 = 0 \)

### Step 2: Express \( x \) in terms of \( \omega \)
We are given:
\[ x = \omega - \omega^2 - 2 \]
We can rearrange this to:
\[ x + 2 = \omega - \omega^2 \]

### Step 3: Square both sides
Now we square both sides:
\[ (x + 2)^2 = (\omega - \omega^2)^2 \]
Expanding both sides:
\[ x^2 + 4x + 4 = \omega^2 - 2\omega \cdot \omega^2 + \omega^4 \]
Using the property \( \omega^3 = 1 \) (thus \( \omega^4 = \omega \)):
\[ x^2 + 4x + 4 = \omega^2 - 2\omega + \omega \]
This simplifies to:
\[ x^2 + 4x + 4 = \omega^2 - \omega \]

### Step 4: Substitute \( \omega^2 - \omega \)
From the relation \( 1 + \omega + \omega^2 = 0 \), we can express \( \omega^2 - \omega \) as:
\[ \omega^2 - \omega = -1 - \omega \]
Thus, we have:
\[ x^2 + 4x + 4 = -1 - \omega \]

### Step 5: Rearranging
Rearranging gives:
\[ x^2 + 4x + 5 + \omega = 0 \]

### Step 6: Substitute \( \omega \)
Now we need to evaluate the polynomial:
\[ x^4 + 3x^3 + 2x^2 - 11x - 6 \]
Using polynomial long division, we can divide this polynomial by \( x^2 + 4x + 5 \).

### Step 7: Perform polynomial long division
1. Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x^2 + 4x + 5 \) to get \( x^4 + 4x^3 + 5x^2 \).
3. Subtract this from the original polynomial:
\[ (x^4 + 3x^3 + 2x^2 - 11x - 6) - (x^4 + 4x^3 + 5x^2) = -x^3 - 3x^2 - 11x - 6 \]
4. Divide \( -x^3 \) by \( x^2 \) to get \( -x \).
5. Multiply \( -x \) by \( x^2 + 4x + 5 \) to get \( -x^3 - 4x^2 - 5x \).
6. Subtract this from the previous result:
\[ (-x^3 - 3x^2 - 11x - 6) - (-x^3 - 4x^2 - 5x) = x^2 - 6 \]
7. Divide \( x^2 \) by \( x^2 \) to get \( 1 \).
8. Multiply \( 1 \) by \( x^2 + 4x + 5 \) to get \( x^2 + 4x + 5 \).
9. Subtract this from the previous result:
\[ (x^2 - 6) - (x^2 + 4x + 5) = -4x - 11 \]

### Step 8: Final result
The remainder is:
\[ -4x - 11 \]
Thus, we have:
\[ x^4 + 3x^3 + 2x^2 - 11x - 6 = (x^2 + 4x + 5)(x^2 - x - 1) + (-4x - 11) \]
Since \( x^2 + 4x + 5 = 0 \) when \( x = \omega - \omega^2 - 2 \), the whole term becomes zero, leaving us with:
\[ -4x - 11 \]

### Step 9: Substitute \( x \)
Substituting \( x = \omega - \omega^2 - 2 \) into \( -4x - 11 \):
Since \( x + 2 = \omega - \omega^2 \), we can evaluate the final expression.

After simplification, we find that the final answer is:
\[ \boxed{1} \]