I understand Hensel's Lemma and how you lift a root $f(b_{j}) \equiv 0 \pmod{p^\alpha}$ to one $f(b_{j+1}) \equiv 0 \pmod{p^{\alpha+1}}$, as long as $f'(b_{j}) \not\equiv 0 \pmod{p}$.
The equation is actually quite simple: $b_{j+1} \equiv b_{j} - f(b)*(f'(a)^{-1}) \pmod{p^{\alpha+1}}$.
My question is what happens when $f'(b) \equiv 0 \pmod{p}$.
The best I can find is that there are multiple solutions, but nothing I can find says how to find them. Is there a nice equation for $b_{j+1}$ (or the multiple values it can take on) given $b_{j}$.