Invertibility depending on a parameter

Row operation. $L_1\leftarrow L_1+L_2+L_3$:
$$\det A_m = \begin{vmatrix} m+2 & m+2 & m+2 \\ 1 & m & 1 \\ 1 & 1 & m \end{vmatrix} = (m+2)\begin{vmatrix} 1 & 1 & 1 \\ 1 & m & 1 \\ 1 & 1 & m \end{vmatrix}.$$

Column operations. $C_2\leftarrow C_2-C_1$, $C_3\leftarrow C_3-C_1$:
$$\det A_m = (m+2)\begin{vmatrix} 1 & 0 & 0 \\ 1 & m-1 & 0 \\ 1 & 0 & m-1 \end{vmatrix} = (m+2)(m-1)^2.$$

Conclusion. $A_m$ is invertible iff $m\neq -2$ and $m\neq 1$.