Eisenbud states the following corollary to Hensel's lemma:
Given a polynomial $f(t,x)$ over a field $k$, with $x=a$ a simple root of $f(0,x)$, then there exists a unique power series $x(t) \in k[[t]]$ with $x(0) = a$ and $f(t,x(t)) = 0$ identically. His hint is to use the complete local ring $R = k[[t]]$ for $R[x]$.
Since $a$ is a simple root of $f(0,a)$, we know that $f'(t,a) \neq 0$. But, I'm not sure why $$ \begin{split} f(t,a) &\equiv 0 \pmod{f'(a)\cdot (t)} \\ &= 0 \pmod{\alpha\cdot(t)} \end{split} $$ This would hold of course if $f(t,a) \equiv f(0,a)$, but I'm not sure this is the case.