I am trying to gain some insight into how the irreducibilty/reducibility of the vanishing set of a homogeneous polynomial affects the composition of the polynomial itself, so I am looking for two distinct examples of a polynomial $g\in\mathbb{C}[x,y,z]$ where $g$ is degree 2 and homogeneous and contains the point $(0:0:1)$ with a multiplicity greater than 1 and such that the same point $(0:0:1)$ is contained in the vanishing set $\mathbb{V}(f)\subset\mathbb{CP}^2$.
For one example, I would like $\mathbb{V}(f)$ to be reducible and for the other, it should be irreducible.