The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2i\\[1 em]x_3 &= -2i \end{aligned} $$Step 1:
Factor out $ \color{blue}{ 2x }$ from $ 2x^3+8x $ and solve two separate equations:
$$ \begin{aligned} 2x^3+8x & = 0\\[1 em] \color{blue}{ 2x }\cdot ( x^2+4 ) & = 0 \\[1 em] \color{blue}{ 2x = 0} ~~ \text{or} ~~ x^2+4 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 2:
The solutions of $ x^2+4 = 0 $ are: $ x = 2 i ~ \text{and} ~ x = -2 i $.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.