The quintic $(x^2+y^2)(x^3-3xy^2)-3x^2y+y^3$ equals $\operatorname{Re}\bigl(z^3(|z|^2+i)\bigr)$

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The quintic $(x^2+y^2)(x^3-3xy^2)-3x^2y+y^3$ equals $\operatorname{Re}\bigl(z^3(|z|^2+i)\bigr)$ · Prove2Me