Answer
No solution in $ \mathbb{Z}_{6}$.
Work Step by Step
The elements of $\;\mathbb{Z}_{6}\;$ are 0,1,2,3,4,5
So, if there is a solution for this equation, then x is one of these numbers.
$If \;\;x=0\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;4.(0)+5=2 \; {\color{Red}\rightarrow } \;\; 0+5= 2\;\;{\color{Red} \Rightarrow }\;\;5\neq 2\\
so \;\;0 \;is \;not \;solution\\\\
If \;x=1\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;4.(1)+5=2 \; {\color{Red}\rightarrow } \;\; 4+5= 2\;\;{\color{Red} \Rightarrow }\;\;3\neq 2\\
so\; 1\; is\; solution.\\\\
If\;x=2\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;4.(2)+5=2 \; {\color{Red}\rightarrow } \;\; 2+5= 2\;\;{\color{Red} \Rightarrow }\;\;1\neq 2\\
so\; 2\; is\;not\; solution\\\\
If \;x=3\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;4.(3)+5=2 \; {\color{Red}\rightarrow } \;\; 0+5= 2\;\;{\color{Red} \Rightarrow }\;\;5\neq 2\\
so\; 3\; is\; not\; solution\\\\
If\; x=4\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;4.(4)+5=2 \; {\color{Red}\rightarrow } \;\; 4+5= 2\;\;{\color{Red} \Rightarrow }\;\;3\neq 2\\
so\; 4 \;is\; not\; solution\\\\
If\; x=5\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;4.(5)+5=2 \; {\color{Red}\rightarrow } \;\; 2+5= 2\;\;{\color{Red} \Rightarrow }\;\;1\neq 2\\
so\; 5 \;is\; solution\\\\$
No solution in $ \mathbb{Z}_{6}$
