<s>[INST] Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution? Does there exist a positive integral solution $(x, y, n)$ to $(xy+1)(xy+x+2)=n^2$? If there doesn't, how does one prove that?
 [/INST] $4n^2=4(xy+1)(xy+x+2)=(3+x+2xy)^2-(1+x)^2$.
For positive even $k$ and $ka^2+b^2=c^2$, when $a,b,c$ is positive integers, and $a=2m,b=km^2-1,c=km^2+1$, when $m$ is any positive integer.
Then $n=2m,1+x=4m^2-1,3+x+2xy=4m^2+1$, and then $2m^2=1$, i.e. $m$ is not integer - contradiction. Means equation $n^2=(xy+1)(xy+x+2)$ has no positive integer solution.
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