On a Hartshorne's statement.

Question

In section 4 of chapter 1 of Hartshorne's Algebraic Geometry it says:

Lemma. Let $Y$ be a hypersurface in $\mathbb{A}^n$ given by equation $f(x_1,\ldots,x_n)=0$. Then $\mathbb{A}^n \setminus Y$ is isomorphic to the hypersurface $x_{n+1}f=1$. In particular, $\mathbb{A}^n \setminus Y$ is affine, and its affine ring is $K[x_1,\ldots,x_n]_f$.

I have two questions:

1. How it follows the statement in blackfold from the previous statement?

2. How can we prove that $H$ is in fact an hypersurface? We know that $f$ is irreducible, and Hartshorne's definition of hypersurface is that is a $V(f)$ with $f$ irreducible, but I don't know how to get that $x_{n+1}f-1$ is irreducible.

Thanks in advance.