prove $xy-zw$ is irreducible in $\mathbb{C}[x,y,z,w]$

Question

In proving this statement why people tend to construct a homomorphism to a domain and conclude $(xy-zw)$ is the kernel and therefore prime? $xy-zw$ is an irrreducible element in $\mathbb C[x,y,z,w]$ . Is the following straightforward arguement wrong? If it is reducible, it must be a product of two homogeneous one degree polynomial，so consider all such possibilities to get a contradiction.Thank you