For some α, β ∈ R, let A = [α, 2; 1, 2] and B = [1, 1; 1, β] be such that A^2-4 A +2 I = B^2-3 B + I = O. Then ( det ( adj ( A^3- B^3)))^2 is equal…

For some $\displaystyle \alpha, \beta \in \mathbf{R}$, let $\displaystyle \mathrm{A}=\left[\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}\right]$ and $\displaystyle \mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 1 & \beta\end{array}\right]$ be such that $\displaystyle \mathrm{A}^2-4 \mathrm{~A}+2 \mathrm{I}=\mathrm{B}^2-3 \mathrm{~B}+\mathrm{I}=\mathrm{O}$. Then $\displaystyle \left(\operatorname{det}\left(\operatorname{adj}\left(\mathrm{A}^3-\mathrm{B}^3\right)\right)\right)^2$ is equal to $\displaystyle \_\_\_\_$.