calculus - What does the group of the geometric images of z, such that $z+overline{z}+|z^2|=0$ define?

What does the group of the geometric images of z, such that
$z+\overline{z}+|z^2|=0$ define?

A) A circumference of center (0,-1) and radius 1.

B) Two lines, of equations $y=x+2$ and $y=x$.

C)A circumference of center (1,0) and radius 1.

D)A circumference of center (-1,0) and radius 1.

I tried to simplify the expression:

$$z+\overline{z}+|z^2|=0 \Leftrightarrow \\ x+yi+x-yi+|(x+yi)^2|=0\Leftrightarrow \\ 2x+\sqrt{(x^2+y^2)^2+(2xy)^2}=0 \Leftrightarrow \\ \sqrt{(x^2+y^2)^2+(2xy)^2}=-2x \Leftrightarrow \\ (x^2+y^2)^2+(2xy)^2=4x^2 \Leftrightarrow \\ x^4 - 2 x^2 y^2 + y^4+4x^2y^2 = 4x^2 \Leftrightarrow \\ x^4+y^4+2x^2y^2 = 4x^2 \Leftrightarrow \\ ???$$

How do I continue from here?

I have also been thinking that if the sum of those three numbers is zero then they could be the vertices of a triangle. I rewrote the expression:

$$\rho \cdot cis(\theta)+\rho cis(-\theta)+|\rho^2 \cdot cis(2\theta)|=0 \Leftrightarrow \\ \rho \cdot cis(\theta)+\rho cis(-\theta)+\rho^2 =0 \Leftrightarrow \\ \rho(cis(\theta)+cis(-\theta)+\rho) = 0 \Leftrightarrow \\ \rho = 0 \lor cis(\theta)+cis(-\theta)+\rho = 0 \Leftrightarrow \\ cis(\theta)+cis(-\theta) = -\rho \Leftrightarrow \\ \cos(\theta)+\sin(\theta)i+\cos(-\theta)+\sin(-\theta)i = -\rho \Leftrightarrow \\ \cos(\theta)+\cos(\theta) = -\rho \Leftrightarrow \\ 2\cos(\theta) = -\rho \Leftrightarrow \\ \rho = -2\cos(\theta)$$

This means that $\rho$ will be between 0 and 2. If $|z^2|=\rho^2 = \rho^2 cis(0)$, then one of the vertices is $4$.

But what do I do next? How do I solve this?

Answer

That solution is very laboured: $|z^2|=x^2+y^2$ so the curve's equation is
$$x^2+y^2+2x=0$$
or

$$(x+1)^2+y^2=1.$$
I'm sure you can identify the curve now.