Answer
$r=4\cos(\theta-\pi)$
or
$ r=-4\cos\theta$
Work Step by Step
The circle passes through the origin, as (0,0) satisfies the Cartesian equation.
The radius is $2$, and the center is at $(-2,0)$.
In polar coordinates, the center lies at $r_{0}=2, \theta_{0}=\pi,\qquad P(2,\pi)$
A circle passing through the origin, of radius $a$, centered at $P_{0}(r_{0}, \theta_{0}),$
has the polar equation
$r=2a\cos(\theta-\theta_{0})$
So this circle has equation
$r=2(2)\cos(\theta-\pi)$
or
$r=4\cos(\theta-\pi)$
We can apply the identity
$\cos(\theta-\pi)=-\cos\theta$,
in which case the equation can also be
$ r=-4\cos\theta$
