$C_{\Bbb R}A = R\backslash A = [m+\infty)$
$C_{\Bbb R}A \cap B \ne \varnothing$
$\Leftrightarrow \left[\begin{array}{l}m \leq 2m - 2\\2m - 2 \leq m \leq 2m + 2\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}m \geq 2\\-2 \leq m \leq 2\end{array}\right.$