A hint for bashy and straightforward solution: This equation can be seen as a quadratic equation in $n$.

a quadratic equation in $n$ ; $n^2 - (2 m + 3) n + 2 m^2=0$ $n = 1/2 (-\sqrt(-4 m^2 + 12 m + 9) + 2 m + 3)$ $n = 1/2 (\sqrt(-4 m^2 + 12 m + 9) + 2 m + 3)$ $m = 0, n = 0$ $m = 0, n = 3$ $m = 3, n = 3$ $m = 3, n = 6$