How to show that a parameterization represents a path?

Question

I have the following equation that represents a path $C:y^2=x^3+x^2$ and a line given by the parameterization $r(t)=(t^2-1,t^3-t)$.

I am told that the parameterization represents the path $C$ ,How can one show that this is in fact true?

Answer 1

Hint: $(t^2-1)^3+(t^2-1)^2=t^6-2t^4+t^2=(t^3-t)^2$.