$$\frac{\mathcal{X}^{m+n+2}\times\mathcal{X}^{m+n+2}}{\mathcal{X}^{2(m+n+1)}}=\mathcal{X}^{2}$$
$\left\{\begin{matrix}\\X\neq 0\text{, }&\text{unconditionally}\\X\in \mathrm{C}\text{, }&m=-\left(n+1\right)\end{matrix}\right.$
$\left\{\begin{matrix}\\m=-\left(n+1\right)\text{, }&\text{unconditionally}\\m\in \mathrm{C}\text{, }&X\neq 0\end{matrix}\right.$
$\left\{\begin{matrix}\\X>0\text{, }&\text{unconditionally}\\X<0\text{, }&Denominator(m+n)\text{bmod}2=1\text{ and }Denominator(2m+2n)\text{bmod}2=1\end{matrix}\right.$
