The polynomial defines a genus 2 curve. Your curve is birational to the genus two curve $C$ given by $y^2=x^5+7x^4+18x^3+24x^2+16x+4$. This curves has two Weierstrass points. These points correspond with the two points at infinity on your curve.

There are standard techniques to bound the number of rational points on such curves. E.g., you can consider first the Jacobian of this curve and try to find a basis for the Mordell-Weil group. If this group is small enough you may be able to find all rational points on $C$. On the web page of Michael Stoll you can find many papers and slides on this subject.

The polynomial defines a genus 2 curve. Your curve is birational to the genus two curve $C$ given by $y^2=x^5+7x^4+18x^3+24x^2+16x+4$. This curves has two rational Weierstrass points. These points correspond with the two points at infinity on your curve.

There are standard techniques to bound the number of rational points on such curves. E.g., you can consider first the Jacobian of this curve and try to find a basis for the Mordell-Weil group. If this group is small enough you may be able to find all rational points on $C$. On the web page of Michael Stoll you can find many papers and slides on this subject.