Question:
Find the sum of all complex numbers $z$ that satisfy
\[z^3 + z^2 - |z|^2 + 2z = 0.\]

Answer:
Since $|z|^2 = z \overline{z},$ we can write
\[z^3 + z^2 - z \overline{z} + 2z = 0.\]Then
\[z (z^2 + z - \overline{z} + 2) = 0.\]So, $z = 0$ or $z^2 + z - \overline{z} + 2 = 0.$

Let $z = x + yi,$ where $x$ and $y$ are real numbers.  Then
\[(x + yi)^2 + (x + yi) - (x - yi) + 2 = 0,\]which expands as
\[x^2 + 2xyi - y^2 + 2yi + 2 = 0.\]Equating real and imaginary parts, we get $x^2 - y^2 + 2 = 0$ and $2xy + 2y = 0.$  Then $2y(x + 1) = 0,$ so either $x = -1$ or $y = 0.$

If $x = -1,$ then $1 - y^2 + 2 = 0,$ so $y = \pm \sqrt{3}.$  If $y = 0,$ then $x^2 + 2 = 0,$ which has no solutions.

Therefore, the solutions in $z$ are 0, $-1 + i \sqrt{3},$ and $-1 - i \sqrt{3},$ and their sum is $\boxed{-2}.$