= Solution
Retain the order of the noncommuting <position operator> and <momentum operator>. Using their <commutator> $[\hat x,\hat p]=i\hbar$, expansion gives
$$
\begin{aligned}
\hat B\hat A&=\hat p^2+m^2\omega^2\hat x^2+im\omega[\hat x,\hat p]
=2m\hat H-m\hbar\omega,\\
\hat A\hat B&=\hat p^2+m^2\omega^2\hat x^2-im\omega[\hat x,\hat p]
=2m\hat H+m\hbar\omega.
\end{aligned}
$$
Thus \b[the two factorizations of the <quantum harmonic oscillator> are]
$$
\boxed{\hat B\hat A=2m(\hat H-\tfrac12\hbar\omega),\qquad
\hat A\hat B=2m(\hat H+\tfrac12\hbar\omega).}
$$
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