The solution set incorporates the straight lines:

x = 0

y = 0

y = -x

y = -x/2

y = -2x

These five lines divide the infinite pie into 10 slices (regions).

Inexplicable is Matlab's failure to show y = -x as one of the lines in doing an implicit graph:

Implicit graphing via

f = @(x,y) x.*y.*(x + y).*(y + x).*(x + 2*y).*(y + 2*x);

fimplicit(f)

axis square

grid

where the fimplicit function plots the zeros of the formula,

shows the x-y plane divided by 4 straight lines going through the origin. Two are the x and y axes themselves, as evidenced also by the x and y as factors. The two others extend through the second and fourth quadrants; one has slope -1/2 and the other has slope -2. Since (0,0) satisfies the equation, there is no discontinuity there.

There are 8 regions that the displayed graph has divided the Cartesian plane into.

If the factor (x+y) is removed (since it's really the same as (y+x)), the line y=-x does appear. I don't know why it does not appear when the superfluous factor is included.

In fact if any of the constituent factors is squared, it's corresponding line disappears in the Matlab implicit plot.