Hint for Ring Theory. UFD.

Question

Let $R = \mathbb{Z}[x]$ a UFD. Show that $2$ and $x$ are relatively prime, but there are no elements $f, g \in \mathbb{Z}[x]$ with $1 = 2f + xg$.

The first affirmation is simpl, but I couldnt conclude the second. Any hint? I didn't want the solution, just a little hint.

Answer 1

Hint: Show that $(2f+xg)(0)\neq 1(0)$ where $1(x)$ is the constant map $x\mapsto 1$

Hint #2: What is the parity of $(2f+xg)(0)$ ?

Answer 2

What do you get when you evaluate $2f+xg$ at $x=0$? You get an integer, for sure. Can you say whether it's even or odd?