Solution
= Solution
Factor the expression using the <complex conjugate>:
$$
\begin{aligned}
z^3-z^2\overline z-z\overline z^2+\overline z^3
&=(z+\overline z)(z-\overline z)^2\\
&=(2x)(2iy)^2=-8xy^2.
\end{aligned}
$$
It is real, so
$$
\boxed{\operatorname{Re}=-8xy^2,
\qquad \operatorname{Im}=0}.
$$