There are various versions of Hensel's lemma in commutative algebra. One is about lifting roots of polynomials (Eisenbud, Theorem 7.3):
Let $R$ be a ring that is complete with respect to the ideal $\mathfrak{m}$ and let $f (x) \in R [x]$ be a polynomial. If $a$ is an approximate root of $f$ in the sense that $$f (a) \equiv 0 \pmod{f' (a)^2 \, \mathfrak{m}}$$ then there if a root $b$ of $f$ near $a$ in the sense that $$f (b) = 0 \quad \text{and} \quad b \equiv a \pmod{f' (a) \, \mathfrak{m}}.$$
And there is also a seemingly more general version about lifting factorizations of polynomials (Eisenbud, Theorem 7.18, Exercise 7.19-7.20):
Let $R$ be a Noetherian ring, complete with respect to an ideal $\mathfrak{m}$. Let $F (x) \in R [x]$ be a polynomial in one variable with coefficients in $R$ and let $f (x)$ be the polynomial over $R/\mathfrak{m}$ obtained by reducing the coefficients of $f$ mod $\mathfrak{m}$. If $f$ factors as $$f = g_1 g_2 \in (R/\mathfrak{m}) [x]$$ in such a way that $g_1$ and $g_2$ generate the unit ideal, and $g_1$ is monic, then there is a unique factorization $$F = G_1\,G_2 \in R [x]$$ such that $G_1$ is monic and $G_i$ reduces to $g_i$ mod $\mathfrak{m}$.
Eisenbud in his textbook gives the second version as an exercise, and explains how to prove it from scratch. He also writes that it can be "deduced in a page" from the first version, but unfortunately, he refers to Nagata's "Local rings" (1962) which I find quite hard to read.
Could anyone explain this implication? It is stated in several places, but I am interested in an intelligible one-page proof, as promised by Eisenbud.
Thank you.