<s>[INST] The graph of the equation $x+y=x^3+y^3$ is the union of The graph of the equation $x+y=x^3+y^3$ is the union of
$(A)$line and an ellipse$(B)$line and a parabola$(C)$line and hyperbola$(D)$line and a point

I tried to factorize the given equation.
$x^3-x+y^3-y=0$
$x(x^2-1)+y(y^2-1)=0$
The answer given is a line and an ellipse but i do not understand how this equation is split into a line and an ellipse equation. 
 [/INST] $$(x+y)=x^3+y^3=(x+y)(x^2-xy+y^2)$$
It can be $\;y=-x\;$ , a straight line, or else, after cancelling:
$$x^2-xy+y^2=1\iff x^2-xy+y^2-1=0\iff$$
$$\left(x-\frac y2\right)^2+\frac34y^2=1\;\;(**)$$
Put now:
$$\begin{cases}u:=x-\frac y2\\{}\\v:=y\end{cases}\iff x=u+\frac v2\;,\;\;y=v$$
and you get the ellipse $\;u^2+\frac34v^2=1\;$
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