The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.56\\[1 em]x_2 &= -1.1143\\[1 em]x_3 &= 1.4264\\[1 em]x_4 &= 0.124+1.3455i\\[1 em]x_5 &= 0.124-1.3455i \end{aligned} $$Polynomial $ 8x^5-24x-13 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.