This looks like an ellipse. We find out by completing the squares for x and y:
12x2+24x+20y2+40y=68,
12(x2+2x+1-1)+20(y2+2y+1-1)=68,
12(x+1)2-12+20(y+1)2-20=68,
12(x+1)2+20(y+1)2=68+12+20=100.
Now divide through by 100:
3(x+1)2/25+(y+1)2/5=1. This is the standard form for the equation of an ellipse.