Question

x is a unit in R[x]

Answer 1

It is false.

$\deg (xp(x))\ge 1$ for any $p(x)\in R[x]\setminus\{0\}$, but $\deg 1=0$. So $xp(x)\neq 1$ for every $p(x)\in R[x]$. That is, $x$ is not unit in $R[x]$ .