I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too.
I can guess that the curve is a polynomial equation so in principle I can write this as
$$w_1 x^4 +w_2 x^3 +w_3 x^2 +w_4 x +w_5 y^3 +w_6 y^2 +w_7 y +w_8=0$$
for the powers of $t$ to cancel when plugged in I am determine all these $w's$ by substituting the $t's$ for $x(t)$ and $y(t)$ and setting each of the new coefficients of the $t$'s $C_i(w_1,...w_8)$ to zero individually. I have my curve determined if I put each of the $C_i(w_1,...w_8)=0$ and solve for the $w$'s.
All this was in principle, but this is like an examination question so there must be a clever manipulation/way, which I have been unable to find.