If you add and subtract you get . Let u= x+y and v= y. x= u- y= u- v so the equation becomes .
For x= u- v>0, |x|= |u-v|= u- v so that becomes . Complete the square by adding 400 and subtracting 400 on the left: . That is a hyperbola, in the uv-plane, with center at (-20, -20), vertices (0, -20) and (-40, -20), and vertical asymptotes at 45 degree angles to the axes. That is, one of the asymptotes is the line u= v and, since we require that u- v> 0 here, we have only the branch of the hyperbola on that side.
If u- v< 0, then |u- v|= v- u> 0 and the equation becomes and, after completing the square, , a hyperbola with center at (20, 20), vertices (0, 20) and (40, 20), and u= v and u= -v as asymptotes again. Since u< v now, we have the branch of the hyperbola now on the opposite side. Those don't cross: one lies on one side of v= u and the other on the opposite side, so I don't see how they can create a bounded region!