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.