The basic idea behind the factorization is to replace a 2nd-order differential equation by a pair of first order ones. It was made popular in physics by the work of Hull and Infeld:
Infeld, Leopold, and T. E. Hull. "The factorization method." Reviews of modern Physics 23.1 (1951): 21
although in fact earlier examples, such as the factorization of the harmonic oscillator into creation and destruction operators, were known before the aforementioned paper. The review paper is apparently openly accessible if you follow the GoogleScholar link
The method is actually quite general and Infeld and Hull do a good job of showing how one can factorize some 2nd-order differential equations.
It is at the core of supersymmetric quantum mechanics, where one finds operators $\hat A^\dagger$ and $\hat A$ so that $\hat A^\dagger \hat A$ and $\hat A\hat A^\dagger$ give two different Hamiltonians, connected by a superpotential. Suppose
\begin{align}
\hat H\psi(x)=\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V_-(x)\right)\psi_0(x)=0\, , \qquad V_-(x)=V(x)-E_0
\end{align}
One can easily show that the Hamiltonian can be rewritten in the form
\begin{align}
\hat H_-=\frac{\hbar^2}{2m}\left(\frac{d^2}{dx^2}+\frac{\psi_0^{''}(x)}{\psi_0(x)}\right)=\hat A^\dagger\hat A\, ,
\end{align}
where
\begin{align}
\hat A=\frac{\hbar}{\sqrt{2m}}\left(\frac{d}{dx}-\frac{\psi^{'}_0(x)}{\psi_0(x)}\right)\, ,\qquad
\hat A^\dagger=\frac{\hbar}{\sqrt{2m}}\left(-\frac{d}{dx}-\frac{\psi^{'}_0(x)}{\psi_0(x)}\right)\, .
\end{align}
The interesting "supersymmetric" bit is that $\hat A\hat A^\dagger$ is the Hamiltonian for a different potential.
More details can be found on this in the review paper by Fred Cooper et al (e-copy here):
Cooper, Fred, Avinash Khare, and Uday Sukhatme. "Supersymmetry and quantum mechanics." Physics Reports 251.5-6 (1995): 267-385.
