$$1-x^{2}-2xy-y^{2}$$
$\left(1-y-x\right)\left(x+y+1\right)$
$$1-\left(x^{2}+2xy+y^{2}\right)$$
$$1^{2}-\left(x+y\right)^{2}$$
$$\left(1-x-y\right)\left(1+x+y\right)$$
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$-\left(x+y\right)^{2}+1$