Past exam of the mathematical methods I course of the University of Oxford / 2019 / 9 / c

By Vieta's formulas, the sum of all five polynomial roots, counted with multiplicity of a root, is minus the coefficient of $z^4$ divided by the leading coefficient. Hence

For a monic polynomial of degree five, Vieta's formulas give the product of its polynomial roots as $(-1{)}^5$ times its constant coefficient. Therefore

The coefficients are real, so the supplied polynomial root $2+2i$ has its complex conjugate $2-2i$ as another polynomial root. Their monic polynomial factor of degree two isPolynomial division then givesThus all five polynomial roots, with multiplicity of a root, areTheir sum is $5$ and their product is $8\cdot 1\cdot (-1)=-8$, agreeing with solution and solution.