If `x+ (1)/(x)= -1` find
`x^(51) + x^(45) + x^(21) + x^(15) + x^(3) + 2`= ?

Answer

To solve the equation \( x + \frac{1}{x} = -1 \) and find the value of \( x^{51} + x^{45} + x^{21} + x^{15} + x^{3} + 2 \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ x + \frac{1}{x} = -1 \]
Multiplying both sides by \( x \) gives:
\[ x^2 + 1 = -x \]
Rearranging this, we have:
\[ x^2 + x + 1 = 0 \]

### Step 2: Solve the quadratic equation
To solve the quadratic equation \( x^2 + x + 1 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 3: Find \( x^3 \)
Next, we can find \( x^3 \) using the fact that \( x^2 + x + 1 = 0 \):
\[ x^3 = - (x^2 + x) = -(-1) = 1 \]
Thus, we have:
\[ x^3 = 1 \]

### Step 4: Reduce the powers of \( x \)
Since \( x^3 = 1 \), we can reduce the powers of \( x \) in the expression \( x^{51} + x^{45} + x^{21} + x^{15} + x^{3} + 2 \):
- \( x^{51} = (x^3)^{17} = 1^{17} = 1 \)
- \( x^{45} = (x^3)^{15} = 1^{15} = 1 \)
- \( x^{21} = (x^3)^{7} = 1^{7} = 1 \)
- \( x^{15} = (x^3)^{5} = 1^{5} = 1 \)
- \( x^{3} = 1 \)

### Step 5: Substitute back into the expression
Now substituting these values back into the expression:
\[ x^{51} + x^{45} + x^{21} + x^{15} + x^{3} + 2 = 1 + 1 + 1 + 1 + 1 + 2 \]
Calculating this gives:
\[ 1 + 1 + 1 + 1 + 1 + 2 = 7 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{7} \]