The values of $x$ for which the given matrix $\left[\begin{array}{ccc}-x& x& 2\\ 2& x& -x\\ x& -2& -x\end{array}\right]$ will be non-singular are:

A matrix is non-singular if its determinant is non-zero ($\det (A)\ne 0$).Let $\Delta =\left|\begin{array}{ccc}-x& x& 2\\ 2& x& -x\\ x& -2& -x\end{array}\right|$.Expanding:
$$\Delta =-x(-x^2-2x)-x(-2x+x^2)+2(-4-x^2)$$
$$\Delta =x^3+2x^2+2x^2-x^3-8-2x^2$$
$$\Delta =2x^2-8$$
For non-singularity:
$$2x^2-8\ne 0$$
$$2x^2\ne 8⟹x^2\ne 4⟹x\ne \pm 2$$
Thus, $x$ can be any value except $2$ and $-2$.