I'm having trouble finding a parameterization for the following curve: $x^4 - 2x^2yz + y^2z^2 - y^3z = 0$ taken to be a curve in $\mathbb{C}\mathbb{P}^2$. I followed the example on Wikipedia where they parameterized a circle's equaton, ie., I demohogenized the polynomial at $z$, so I reduce to the curve $x^4 - 2x^2y + y^2 - y^3 = 0$. Then I observed that the curve contains the point $(0,1)$, so I considered the line through $(0,1)$ with slope $t$, ie. $y = tx + 1$ and I plugged this into my equation.
I ended up with an ugly quartic polynomial in $x$ with coefficients in $t$...Specifically, I got $x^4 -2tx^3 + (2+t^2)x^2 + (-t-2t)x + 2 = 0$ and hopefully I didn't make any mistakes. Solving for $x$ doesn't seem easy. Is this the right approach to doing this?