This is an exercise from Introduction to Plane Algebraic Curves by Kunz:
Let $c:\mathbb{P}^2(K) \rightarrow \mathbb{P}^2(K)$ be the coordinate transformation that sends points $[1:0:0],[0:1:0],[-1:0:1],[0,1,1]$
to points
$[1:0:0],[0:1:0],[0:0:1],[1,1,1]$.
a)Determine the equation of the curve $$X_0^2X_2-X_0X_1^2+X_0X_2^2-2X_0X_1X_2-X_1^2X_2-2X_1X_2^2$$ in the new coordinate system.
b) Is this curve irreducible?
So I think in order to do b) I need to use the form of curve from a). But I am not sure how to do this, the question gives the transformation of 4 points, but I think we only need 3 of them. Also after I get the transformation matrix
$$\begin{bmatrix} 1 & 0 & 1\\0&1&0\\0&0&1 \end{bmatrix} $$
How should I apply this to the curve?