Given that ` 1 + 2|z|^(2) = |z^(2) + 1|^(2) + 2 | z + 1 | ^(2)`, then the value of `|z(z + 1 )|` is ______.

Explanation

To solve the problem, we start with the given equation:

\[ 1 + 2|z|^2 = |z^2 + 1|^2 + 2|z + 1|^2 \]

### Step 1: Rewrite the equation using properties of complex numbers
We know that \( |z|^2 = z \cdot \overline{z} \). Therefore, we can rewrite the equation as:

\[ 1 + 2(z \cdot \overline{z}) = |z^2 + 1|^2 + 2|z + 1|^2 \]

### Step 2: Expand the right-hand side
Next, we need to expand \( |z^2 + 1|^2 \) and \( |z + 1|^2 \):

1. \( |z^2 + 1|^2 = (z^2 + 1)(\overline{z^2} + 1) = (z^2 + 1)(\overline{z}^2 + 1) \)
2. \( |z + 1|^2 = (z + 1)(\overline{z} + 1) \)

### Step 3: Substitute back into the equation
Now, substituting these expansions back into the equation gives us:

\[ 1 + 2(z \cdot \overline{z}) = (z^2 + 1)(\overline{z}^2 + 1) + 2(z + 1)(\overline{z} + 1) \]

### Step 4: Simplify the equation
Now we simplify the right-hand side:

1. \( (z^2 + 1)(\overline{z}^2 + 1) = z^2\overline{z}^2 + z^2 + \overline{z}^2 + 1 \)
2. \( 2(z + 1)(\overline{z} + 1) = 2(z\overline{z} + z + \overline{z} + 1) = 2(z\overline{z}) + 2z + 2\overline{z} + 2 \)

Combining these gives us:

\[ 1 + 2(z \cdot \overline{z}) = z^2\overline{z}^2 + z^2 + \overline{z}^2 + 1 + 2(z\overline{z}) + 2z + 2\overline{z} + 2 \]

### Step 5: Collect like terms
Collecting like terms leads to:

\[ 0 = z^2\overline{z}^2 + z^2 + \overline{z}^2 + 2 + 2(z\overline{z}) + 2z + 2\overline{z} - 2(z\overline{z}) - 1 \]

### Step 6: Set up the equations
From the simplification, we can derive two equations:

1. \( z \cdot \overline{z} = 1 \)
2. \( z + \overline{z} = -1 \)

### Step 7: Solve for \( z \)
From \( z \cdot \overline{z} = 1 \), we have \( |z| = 1 \). The second equation can be rewritten as:

\[ z + \frac{1}{z} = -1 \]

Multiplying through by \( z \) gives:

\[ z^2 + z + 1 = 0 \]

### Step 8: Find the roots
The roots of this equation can be found using the quadratic formula:

\[ z = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 9: Calculate \( |z(z + 1)| \)
Now, we need to find \( |z(z + 1)| \):

1. \( z + 1 = \frac{-1 \pm i\sqrt{3}}{2} + 1 = \frac{1 \pm i\sqrt{3}}{2} \)
2. Therefore, \( |z(z + 1)| = |z| \cdot |z + 1| = 1 \cdot \left| \frac{1 \pm i\sqrt{3}}{2} \right| = \frac{1}{2} \sqrt{1^2 + (\sqrt{3})^2} = \frac{1}{2} \cdot 2 = 1 \)

Thus, the value of \( |z(z + 1)| \) is:

\[ \boxed{1} \]