Treat the division as a fraction, simplify, and then rationalize the denominator.

$\displaystyle\frac{{{8}-{2}{i}}}{{{2}{i}}}=\frac{{2}}{{2}}\cdot\frac{{{\left({4}-{i}\right)}}}{{i}}=\frac{{{4}-{i}}}{{i}}\cdot\frac{{i}}{{i}}=\frac{{{4}{i}-{{i}}^{{2}}}}{{{{i}}^{{2}}}}=\frac{{{4}{i}-{\left(-{1}\right)}}}{{-{1}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{4}{i}+{1}}}{{-{1}}}=-{4}{i}-{1}=-{1}-{4}{i}\usepackage{cmbright}\Huge\$