Let ` omega ne 1 ` be a complex cube root of unity. If
` ( 4 + 5 omega + 6 omega ^(2)) ^(n^(2) + 2) + ( 6 + 5omega^(2) + 4 omega ) ^(n ^(2) + 2) + ( 5+ 6 omega + 4 omega ^(2) ) ^( n ^(2) + 2 ) = 0`, and ` n in N ` , where `n in [1, 100]`, then number of values of n is _______.

Explanation

To solve the problem, we need to analyze the given equation involving complex cube roots of unity. Let's break it down step by step.

### Step 1: Understanding the Complex Cube Roots of Unity
The complex cube roots of unity are given by:
- \( \omega = e^{2\pi i / 3} \)
- \( \omega^2 = e^{-2\pi i / 3} \)
- \( \omega^3 = 1 \)

Given that \( \omega \neq 1 \), we know that \( 1 + \omega + \omega^2 = 0 \).

### Step 2: Simplifying the Expression
The equation we need to evaluate is:
\[ (4 + 5\omega + 6\omega^2)^{(n^2 + 2)} + (6 + 5\omega^2 + 4\omega)^{(n^2 + 2)} + (5 + 6\omega + 4\omega^2)^{(n^2 + 2)} = 0 \]

### Step 3: Identifying Symmetry
Notice that the three terms in the equation can be rewritten using the properties of \( \omega \):
1. \( 4 + 5\omega + 6\omega^2 \)
2. \( 6 + 5\omega^2 + 4\omega \)
3. \( 5 + 6\omega + 4\omega^2 \)

We can factor out common terms:
- Let \( A = 4 + 5\omega + 6\omega^2 \)
- Let \( B = 6 + 5\omega^2 + 4\omega \)
- Let \( C = 5 + 6\omega + 4\omega^2 \)

### Step 4: Using Properties of Roots of Unity
By substituting \( \omega^2 = -1 - \omega \) into \( A \), \( B \), and \( C \), we can find their values:
- \( A = 4 + 5\omega + 6(-1 - \omega) = 4 + 5\omega - 6 - 6\omega = -2 - \omega \)
- \( B = 6 + 5(-1 - \omega) + 4\omega = 6 - 5 - 5\omega + 4\omega = 1 - \omega \)
- \( C = 5 + 6\omega + 4(-1 - \omega) = 5 + 6\omega - 4 - 4\omega = 1 + 2\omega \)

### Step 5: Setting Up the Equation
Now we can rewrite the equation:
\[ (-2 - \omega)^{(n^2 + 2)} + (1 - \omega)^{(n^2 + 2)} + (1 + 2\omega)^{(n^2 + 2)} = 0 \]

### Step 6: Finding Conditions for Zero
For the sum to equal zero, we need to analyze the conditions under which these powers can sum to zero. This typically occurs when the magnitudes of the terms are equal and they are symmetrically placed in the complex plane.

### Step 7: Solving for \( n \)
To satisfy the equation, we can set:
\[ n^2 + 2 \equiv 0 \mod 3 \]
This means \( n^2 \equiv 1 \mod 3 \). The values of \( n \) that satisfy this condition are:
- \( n \equiv 1 \mod 3 \) or \( n \equiv 2 \mod 3 \).

### Step 8: Counting Valid \( n \)
The values of \( n \) in the range \( [1, 100] \) that satisfy \( n \equiv 1 \mod 3 \) are:
1, 4, 7, ..., 100 (33 terms).

The values of \( n \) that satisfy \( n \equiv 2 \mod 3 \) are:
2, 5, 8, ..., 98 (33 terms).

### Final Count
Adding both sets gives us \( 33 + 33 = 66 \) valid values of \( n \).

### Conclusion
Thus, the total number of values of \( n \) for which the given equation holds true is:
\[ \boxed{66} \]