Factorize $xy(xy + 1) + (xy + 3) - 2(x + y + 1/2) - (x + y - 1)^2$ Factorize $xy(xy + 1) + (xy + 3) - 2(x + y + \frac{1}{2}) - (x + y - 1)^2$
When I substitute $x = 1$, $x = -1$, $y = 1$, $y = -1$, It gives a result of $0$. So the result will be $(x+1)(x-1)(y+1)(y-1)$ after checking.
However, it seems that the question pack these terms in a order for a reason(i.e. doesn't require student to expand). So what is the smart way to do it?
Any hints are welcomed. Thanks in advance.

$$xy(xy+1)+(xy+3)-2(x+y+1/2)-(x+y-1)^2$$
$$xy(xy+1)+(xy+1)+2-2(x+y+1/2)-(x+y-1)^2$$
$$(xy+1)^2+2-2(x+y+1/2)-(x+y-1)^2$$
$$(xy+1)^2+2-2x-2y-1-(x+y-1)^2$$
$$(xy+1)^2-2(x+y-1)-1-(x+y-1)^2$$
$$(xy+1)^2-1-(x+y-1)(x+y+1)$$
$$(xy+1)^2-1-((x+y)^2-1)$$
$$(xy+1)^2-(x+y)^2$$
$$\Big[xy+1-(x+y)\Big]\Big[xy+1+(x+y)\Big]$$
$$\Big[y(x-1)+(1-x)\Big]\Big[y(x+1)+(1+x)\Big]$$
$$(x-1)(y-1)(x+1)(y+1)$$
