Consider the wave equation $$\frac{\partial^2u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0\tag{1}$$ which can be written as $$\Big(\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}\Big)\Big(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\Big)u=0\tag{2}.$$
The question here claims that (2) implies both $\Big(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\Big)u=0$ as well as $\Big(\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}\Big)u=0$.
I do not get this. Eq. (2) can be satisfied if $\Big(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\Big)u=v\neq 0$ and $$\Big(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\Big)v=0.$$
Am I missing something?