Find all $(m,x)\in\mathbb{N},$ such that $m^x +x=m^2 +x^2$
My try: The equation $m^x +x=m^2 +x^2$ can be further written as $m^2 (m^{x-2}-1)=x(x-1).$ If $x=3\implies m^2 (m-1)=6\implies\text{No solution}.$
If $x=4\implies m^2 (m-1)=12\implies m=2.$
Now,since, $m^2$ and $(m^{x-2}-1)$ are relatively prime,as are $x,(x-1)$
But I don't know how to proceed from here. Any help would be appreciated.Thank you!