2016 AMC 12A Problem 7

Moving all terms to one side gives $$(x^2-y^2)(x+y+1)=0,$$ which factors as $$(x-y)(x+y)(x+y+1)=0.$$ The graph is therefore the union of the lines $x=y,$ $x=-y,$ and $x+y+1=0.$ The first two lines intersect at the origin, but the third line $x+y=-1$ is parallel to $x+y=0$ and does not pass through the origin. So the graph consists of three lines that do not all pass through a common point. Thus, the correct answer is D.