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 if and only if $m \neq -2$ and $m \neq 1$.