Recursive Polynomial Help

So, I'm writing a program in which we iterate through a for loop to display the different factors of a polynomial, which I've already finished. However, I was wondering if there was a way to do this recursively? I realize that this is much less efficient, but I'm not that great at recursion and could use the practice.
Not looking so much for code, as much as a logical or theoretical representation of how you would go about solving the problem. Any ideas?
Base case: M(x) = \alpha (x-b)
Recursive step: P(x) = M(x)^p Q(x)
Hopefully I don't sound too dumb here, but would you break that down a little bit more for me, in layman's terms? haha. Not the greatest in mathematical notation :/
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