As I outlined in the same question you crossposted, this is essentially a factorization problem; you're given a polynomial over $GF(2)$, and you want to know the pairs of possible products that make up that polynomial.
This can be reduced to factoring the polynomial into the multiset of prime factors (in the case you listed, the factorization is $73af = 83 \times e5$, where both $83$ and $e5$ are prime polynomials) and then going through the various possible ways of separating out the multiset into two (and multiplying the two).
So, the question is now "how do you factor a polynomial?". A survey of some fast algorithms to do this is here; to start off with, you might want to start with trial division.
That is, to look for prime factors of a polynomial $P$, you scan through small polynomials $Q$ (and you need to scan through only the prime polynomials), and see if $Q$ is a divisor of $P$. You can do that by running the multiplication operation in reverse; you shift $Q$ so that the msbit of $Q$ is the same as the msbit of $P$, exclusive or them, and replace $P$ with that exclusive-or. You repeat until $P$ is smaller than $Q$ (and so you can't continue); if you end up with $P=0$, then $Q$ is a factor (and by keeping track of the shifts, that'll give you the polyonomial $P/Q$).
