Answer
No solution in $\mathbb{Z}_{4}$.
Work Step by Step
The elements of $\mathbb{Z}_{4}$ is 0,1,2,3
So, if there is a solution for this equation, then $x$ is one of these numbers.
$If\;\; x=0\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(0)=1 \;\;\;\;{\color{Red}\rightarrow } \;\;\;\;\; 0\neq 1\\
so\;0 \;is\; not\; solution\\\\\\\\
If \;\;x=1\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(1)=1\;\; {\color{Red}\rightarrow } \;\;\; 2\neq 1\\
so \;1\;\; is\; not\; solution\\\\
If \;x=2\\
\;\;\;\;\;\;\;\;\;\;\;\;\;2.(2)=1 \;\;{\color{Red}\rightarrow }\;\; 4\neq 1\\
so \;2\;\; is\; not\; solution\\\\
If \;x=3\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(3)=1\;\; {\color{Red}\rightarrow } \;\; 6\neq 1\\
so\; 2\;\; is\; not\; solution\\\\$
so, no solution in $\mathbb{Z}_{4}$.
