Find the complex number z for which |z| = z + 1 + 2i.

Explanation

To find the complex number \( z \) such that \( |z| = z + 1 + 2i \), we can follow these steps:

### Step 1: Define the complex number
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

### Step 2: Write the modulus of \( z \)
The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

### Step 3: Set up the equation
According to the problem, we have:
\[ \sqrt{x^2 + y^2} = z + 1 + 2i \]
Substituting \( z = x + iy \) into the equation gives:
\[ \sqrt{x^2 + y^2} = (x + 1) + (y + 2)i \]

### Step 4: Separate real and imaginary parts
From the equation \( \sqrt{x^2 + y^2} = (x + 1) + (y + 2)i \), we can equate the real and imaginary parts:
1. The real part: \( \sqrt{x^2 + y^2} = x + 1 \)
2. The imaginary part: \( 0 = y + 2 \)

### Step 5: Solve for \( y \)
From the imaginary part equation:
\[ y + 2 = 0 \implies y = -2 \]

### Step 6: Substitute \( y \) into the real part equation
Now substitute \( y = -2 \) into the real part equation:
\[ \sqrt{x^2 + (-2)^2} = x + 1 \]
This simplifies to:
\[ \sqrt{x^2 + 4} = x + 1 \]

### Step 7: Square both sides
Squaring both sides to eliminate the square root gives:
\[ x^2 + 4 = (x + 1)^2 \]
Expanding the right side:
\[ x^2 + 4 = x^2 + 2x + 1 \]

### Step 8: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 4 = 2x + 1 \]
Now, isolate \( x \):
\[ 2x = 4 - 1 \implies 2x = 3 \implies x = \frac{3}{2} \]

### Step 9: Write the complex number \( z \)
Now that we have both \( x \) and \( y \):
\[ z = x + iy = \frac{3}{2} - 2i \]

### Final Answer
The complex number \( z \) is:
\[ \boxed{\frac{3}{2} - 2i} \]