Prove that a parametric equation's range is subset of a cartesian equation

$r(t)=t(t-2{)}^{3}i+t(t-2{)}^{2}j$ where $r:\mathbb{R}\to {\mathbb{R}}^{2}$

$C=\{(x,y)\in {\mathbb{R}}^{2}\mid {y}^{4}={x}^{3}+2{x}^{2}y\}$

I have difficulty with this question. This is where I am up to.

Find $dy/dx$ of the cartesian curve $dy/dx=(3{x}^{2}+4xy)/(4{y}^{3}-2{x}^{2})$

$(4{y}^{3}-2{x}^{2})=0,y=(2{x}^{2/3})/2$

Substitute y into cartesian equation and find $x=-27/16,0$

I'm not sure where to go from here, can anyone clarify if I am on the right track, thanks!