Answer
$(x^{2}+y^{2}+x)^{2}=4(x^{2}+y^{2})$
Work Step by Step
Formulas relating rectangular $(x,y)$ and polar $(r,\theta)$ coordinates:
$ x=r\cos\theta$ and $ y=r\sin\theta$
$r^{2}=x^{2}+y^{2}$ and $\displaystyle \tan\theta=\frac{y}{x}$.
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$r=2-\cos\theta|$
... multiply both sides with r,
$ r^{2}=2r-r\cos\theta$
... substitute $r^{2}$ and $ r\cos\theta$ using the formulas
$x^{2}+y^{2}=2r-x\qquad /+x$
$ x^{2}+y^{2}+x=2r\qquad$... square both sides,
$(x^{2}+y^{2}+x)^{2}=4r^{2}\qquad $... substitute $r^{2}$
$(x^{2}+y^{2}+x)^{2}=4(x^{2}+y^{2}),$
... an equation in rectangular coordinates.