Answer
$$
r= -\frac{12}{2+ 3\cos \theta}.
$$
Work Step by Step
The polar equation of the conic of eccentricity $e \gt0$, with focus at the origin and directrix $x = d$ is
$$
r=\frac{e d}{1+e \cos \theta}.
$$
Now, since $e=3/2$ and $x=-4$, then the polar equation is
$$
r=\frac{-6}{1+ (3/2)\cos \theta} =-\frac{12}{2+ 3\cos \theta}.
$$
