展開はしないほうがいいです

\begin{align*} \left(x+y\right)^{2}\left(xy-1\right)+1 & =xy\left(x+y\right)^{2}-\left(x+y\right)\left(x+y\right)+1\\ & =xy\left\{ \left(x+y\right)^{2}-\left(\frac{1}{x}+\frac{1}{y}\right)\left(x+y\right)+\frac{1}{xy}\right\} \\ & =xy\left(x+y-\frac{1}{x}\right)\left(x+y-\frac{1}{y}\right)\\ & =\left(x^{2}+xy-1\right)\left(y^{2}+xy-1\right) \end{align*}