Answer
$16x^2-48y^2-144+192x=0$
Work Step by Step
After multiplying with the denominator of the fraction: $r(4+8\sin{\theta})=12\\4r+8r\sin{\theta}=12\\4r=12-8r\sin{\theta}\\16r^2=(12-8r\sin{\theta})^2$.
We know that $r^2=x^2+y^2$ and that $x=r\cos\theta,y=r\sin\theta$.
Hence $16r^2=(12-8r\sin{\theta})^2$ becomes: $16(x^2+y^2)=(12-8y)^2\\16x^2+16y^2=144-192x+64y^2\\16x^2-48y^2-144+192x=0$
