Prove that the polynomial $P=2X+1 \in \Bbb Z_4[X]$ has an inverse element. What happens if we consider $P$ as an element of $\Bbb Q [X]$?

If $P \in \Bbb Z_4[X]$, then any $Q=kX+1$ where $k \equiv 2 \pmod{4}$ will work as an inverse right? I suppose that the result will not change in $\Bbb Q [X]$ as the coefficients have a multiplicative inverses except $0$?