This implicit equation is simple enough to be converted to explicit equations
eqn = x^2 + y^2 == (2 x^2 + 2 y^2 - x)^2;yExpr = (y /. Solve[eqn, y]);yMax = SortBy[Maximize[#, x] & /@ yExpr, N[First[#]] &][[-1, 1]];yMin = SortBy[Minimize[#, x] & /@ yExpr, N[First[#]] &][[1, 1]];xExpr = (x /. Solve[{eqn, yMin < y < yMax}, x, Reals]) // Simplify[#, yMin < y < yMax] &;xMax = SortBy[Maximize[#, y] & /@ xExpr // Quiet, N[First[#]] &][[-1, 1]];xMin = SortBy[Minimize[#, y] & /@ xExpr, N[First[#]] &][[1, 1]];ContourPlot[ Evaluate[{eqn, x == xMin, x == xMax, y == yMin, y == yMax}], {x, -1, 2}, {y, -1, 1}, AspectRatio -> Automatic]Image may be NSFW.
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