May/June 2026 Paper 31

The square roots of $-71-\left(20\sqrt{3}\right)\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact.

By first forming a quartic equation, find the square roots of $-71-\left(20\sqrt{3}\right)\mathrm{i}$. Give your answer in exact Cartesian form.