[tex]\left|\begin{array}{ccc}m&1&1\\5&-2&1\\m+1&2&3\end{array}\right| \overset{C_3\to(-1)\cdot C_2+C_3}{=}\left|\begin{array}{ccc}m&1&0\\5&-2&3\\m+1&2&1\end{array}\right| = -12 \\ \\ \\\Rightarrow -2m+0+3(m+1)-0- 6m -5= -12\\ \\ \Rightarrow -2m+3m+3-6m-5 = -12 \\ \\ \Rightarrow -5m-2 = -12 \\ \\ \Rightarrow -5m = -10 \\ \\ \Rightarrow \boxed{m = 2}[/tex]
