If $α,β,γ$ are the roots of $x^3 + ax^2 + b = 0$, then the value of $\begin{vmatrix}α& β&γ\\β&γ&α\\γ&α&β\end{vmatrix}$, is 

$a^3$

Since $α,β,γ$ are the roots of the given equation.

$α+β+γ=-a, αβ + βγ + γα = 0$ and $αβγ=-b$.

$Δ=\begin{vmatrix}α& β&γ\\β&γ&α\\γ&α&β\end{vmatrix}$

$Δ=-(α+β+γ)(α^2+β^2+γ^2-αβ -βγ - γα)$

$Δ=-(α+β+γ)\{(α+β+γ)^2-3(αβ + βγ + γα)\}$

$Δ=-(-a)(a^2-0)=a^3$