I'm trying to find a rational parameterization of $$x^2+4xy+4y+2=0$$
We can see that there are no integer points on this curve, but how to prove it?
We can get this form: $(x+1)(x+4y-1)=-3$ and we can analyze the conditions (for example, if $ab=-3$ then $a=3$ and $b=-1$, etc).
Unfortunately, I have a problem finding rational parametrization.
