Find the sum of all complex numbers $z$ that satisfy
\[z^3 + z^2 - |z|^2 + 2z = 0.\]
Question:
Find the sum of all complex numbers $z$ that satisfy
\[z^3 + z^2 - |z|^2 + 2z = 0.\]
Answer:
Let's think step by step.

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