jmc

To deal with the $|x|$ term, we take cases on the sign of $x$:

If $x\ge 0$, then we have $y^2+2xy+40x=400$. Isolating $x$, we have $x(2y+40)=400-y^2$, which we can factor as $$2x(y+20)=(20-y)(y+20).$$Therefore, either $y=-20$, or $2x=20-y$, which is equivalent to $y=20-2x$.

If $x<0$, then we have $y^2+2xy-40x=400$. Again isolating $x$, we have $x(2y-40)=400-y^2$, which we can factor as $$2x(y-20)=(20-y)(y+20).$$Therefore, either $y=20$, or $2x=-y-20$, which is equivalent to $y=-20-2x$.

Putting these four lines together, we find that the bounded region is a parallelogram with vertices at $(0,\pm 20)$, $(20,-20)$, and $(-20,20)$, as shown below: The height of the parallelogram is $40$ and the base is $20$, so the area of the parallelogram is $40\cdot 20=\boxed{800}$.