[2021年福島大学後期・数学第1問]因数分解

\begin{align*} x^{4}+x^{2}+1+2xy-y^{2} & =-\left(y^{2}-2xy-x^{4}-x^{2}-1\right)\cmt{x\text{は4次、}y\text{は2次なので次数の低い}y\text{についてまとめる}}\\ & =-\left(y^{2}-2xy-\left(x^{4}+x^{2}+1\right)\right)\cmt{y\text{の0次の項は複2次式}}\\ & =-\left(y^{2}-2xy-\left(x^{4}+2x^{2}+1-x^{2}\right)\right)\\ & =-\left(y^{2}-2xy-\left(\left(x^{2}+1\right)^{2}-x^{2}\right)\right)\\ & =-\left(y^{2}-2xy-\left(x^{2}+1-x\right)\left(x^{2}+1+x\right)\right)\\ & =-\left(y^{2}-2xy-\left(x^{2}-x+1\right)\left(x^{2}+x+1\right)\right)\\ & =-\left(y+\left(x^{2}+x+1\right)\right)\left(y-\left(x^{2}-x+1\right)\right)\\ & =\left(x^{2}+x+1+y\right)\left(x^{2}-x+1-y\right) \end{align*}