I am trying to understand the proof of Laurent phenomenon of cluster algebras in [the book](https://arxiv.org/abs/1608.05735). On page 45, it is said that "We see the $x_j'$ is linear in $x_q$ and hence irreducible (as a Laurent polynomial in $\tilde{\bf x}$), i.e., it cannot be written as a product of two non-monomial factors." Here \begin{align} x_j' = x_j^{-1} (x_k^c x_q M_1 + M_2), \end{align} where $M_1, M_2$ are monomials in $x_i$'s ($i \not\in \{j,k,q,r\}$). Why $x_j'$ cannot be written as a product of two non-monomial factors. Maybe we can have something like \begin{align} x_j' = (x_q+x_r)(x_j+1)? \end{align} Here $(x_q+x_r)(x_j+1)$ is linear in $x_q$ and $(x_q+x_r)(x_j+1)$ is a product of two non-monomial factors. Where do I made a mistake? Thank you very much.