If `(x+1)^(2) =x`, the value of `11x^(3) + 8x^(2) + 8x -2` is:

Answer

To solve the equation `(x + 1)² = x` and find the value of `11x³ + 8x² + 8x - 2`, we will follow these steps:

### Step 1: Solve the given equation
The first step is to expand the left side of the equation:
\[ (x + 1)² = x \implies x² + 2x + 1 = x \]
Now, rearranging the equation gives:
\[ x² + 2x + 1 - x = 0 \implies x² + x + 1 = 0 \]
This is our equation (1).

### Step 2: Find the roots of the quadratic equation
To find the roots of the quadratic equation \(x² + x + 1 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b² - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 1\), and \(c = 1\):
\[ x = \frac{-1 \pm \sqrt{1² - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]
This means:
\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 3: Substitute the roots into the polynomial
Next, we need to evaluate \(11x³ + 8x² + 8x - 2\). To do this, we will first find \(x²\) and \(x³\) using the equation \(x² + x + 1 = 0\).

From \(x² + x + 1 = 0\), we can express \(x²\) as:
\[ x² = -x - 1 \]

Now, we can find \(x³\) by multiplying \(x²\) by \(x\):
\[ x³ = x \cdot x² = x(-x - 1) = -x² - x \]
Substituting \(x²\) from above:
\[ x³ = -(-x - 1) - x = x + 1 - x = 1 \]

### Step 4: Substitute \(x²\) and \(x³\) into the polynomial
Now we can substitute \(x²\) and \(x³\) into the polynomial:
\[ 11x³ + 8x² + 8x - 2 = 11(1) + 8(-x - 1) + 8x - 2 \]
This simplifies to:
\[ = 11 - 8x - 8 + 8x - 2 \]
The \(8x\) terms cancel out:
\[ = 11 - 8 - 2 = 1 \]

### Final Answer
Thus, the value of \(11x³ + 8x² + 8x - 2\) is:
\[ \boxed{1} \]