Regular expression: not containing string

While Thomas Dickey's answer is clever, there is a right way to do this:

awk '!/domain/ {print}' <<EOF
This is domain test
This is do test
This is test
EOF

This is do test
This is test

[[ ... ]] tokenisation clashes with regular expressions (more on that in my answer to your follow-up question) and \ is overloaded as a shell quoting operator and a regexp operator (with some interference between the two in bash), and even when there's no apparent reason for a clash, the behaviour can be surprising. Rules can be confusing.

Who can tell what these will do without trying it (on all possible input) with any given version of bash?

[[ $a = a|b ]]
[[ $a =~ a|b ]]
[[ $a =~ a&b ]]
[[ $a =~ (a|b) ]]
[[ $a =~ ([)}]*) ]]
[[ $a =~ [/\(] ]]
[[ $a =~ \s+ ]]
[[ $a =~ ( ) ]]
[[ $a =~ [ ] ]]
[[ $a =~ ([ ]) ]]

You can't quote the regexps, because if you do, since bash 3.2 and if bash 3.1 compatibility has not been enabled, quoting the regexps removes the special meaning of RE operator. For instance,

[[ $a =~ 'a|b' ]]

Matches if $a contains a litteral a|b only.

Storing the regexp in a variable avoids all those problems and also makes the code compatible to ksh93 and zsh (provided you limit yourself to POSIX EREs):

regexp='a|b'
[[ $a =~ $regexp ]] # $regexp should *not* be quoted.

There's no ambiguity in the parsing/tokenising of that shell command, and the regexp that is used is the one stored in the variable without any transformation.