Let $m,n\in\mathbb{R}; x_1,x_2,x_3 $ the roots of $x^3+mx+n=0$ and the matrix $A=\begin{pmatrix} 1 & 1 &1 \\ x_1 &x_2 &x_3 \\ x^{2}_1 & x^{2}_2 & x^{2}_3 \end{pmatrix}$
I need to find determinant of $A^2$ which is $det(A)\cdot det(A)$
I got $det(A)=(x_2-x_1)(x_3-x_1)(x_3-x_2)$.I know that $x_1+x_2+x_3=0$, $x_1x_2+x_1x_3+x_2x_3=m$, $x_1x_2x_3=-n$.
I expanded the determinant and I tried to factorize but I can't use Vieta formula because I don't get a sum or a product.
Also I tried to find $det(A\cdot A^T)$ but the calculations are very heavy.
How to approach the exercise?