The real values of a for which the quadratic equation $2x^2-\left(a^3+8a-1\right)x+a^2-4a=0$ possesses roots of opposite signs are given by

The roots of the given equation will be of opposite signs if they are real and their product is negative. 
i.e. Discriminant $\ge 0$ and Product of roots $<0$

$\Rightarrow {\left(a^3+8a-1\right)}^2-8\left(a^2-4a\right)\ge 0$ and $\frac{a^2-4a}{2}<0$

$\begin{array}{l}\Rightarrow a^2-4a<0\left[\because a^2-4a<0\Rightarrow {\left(a^3+8a-1\right)}^2-8\left(a^2-4a\right)\ge 0\right]\\ \Rightarrow 0<a<4\end{array}$