Henselization of the ring of polynomials

Question

I am trying to understand example of Henselization from wiki.

http://en.wikipedia.org/wiki/Henselian_ring#Henselization

It says that Henselization of the ring of polynomials localized at point $(0, \dots, 0)$ are algebraic formal power series (the formal power series satisfying an algebraic equation).

Let call ring of algebraic formal power series by $A_{Hen}$

Question 1: Why this ring is Henselian?

In particular I want to understand why it is local. It seems unclear for me. Consider case of one variable. Consider ideals $(x, 1 \pm \sqrt{1+x})$ in $A_{Hen}$. It seems like these ideals are proper. Consider maximal ideals $ \mathfrak{m}_{ \pm} $ such that $(x, 1 \pm \sqrt{1+x}) \subset \mathfrak{m}_{ \pm} $. Clearly $\mathfrak{m}_{ +} \neq \mathfrak{m}_{ -}$

So $A_{Hen}$ is not local.

Question 2:Where is mistake?