GroebnerBasis[{1/6 (a +2 x)^2 ((a +2 x)^2-12)+4/9 y^2 ((a+2 x)^2+2)+32 y^4/81, a ^3-6 a +8 x^3-4 a x^2-2 (a ^2+2) x}, {x,y},{a}]