jmc

Completing the square in $x$ we can write this as $$(x-3{)}^2-9+2y^2-20y+59=12.$$Next, completing the square in $y$, this becomes $$(x-3{)}^2-9+2(y-5{)}^2-50+59=12.$$Combining all the constants, we have $$(x-3{)}^2+2(y-5{)}^2=12.$$This is the equation of an $\boxed{\text{ellipse}}$.