The roots of polynomial $ p(m) $ are:
$$ \begin{aligned}m_1 &= 2i\\[1 em]m_2 &= -2i\\[1 em]m_3 &= 2i\\[1 em]m_4 &= -2i \end{aligned} $$Polynomial $ m^4+8m^2+16 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.