You first example will create a string, not a regular expression object.
var myStr = 'test';
var regEx = '/'+myStr+'/i';
console.log(typeof regEx);//string
Using RegExp will create a regular expression object.
var myStr = 'test';
var regEx = RegExp(myStr,'i');
console.log(typeof regEx);//object
Thus when you try to use the search method, you are searching with a string on slashes on both sides, thus getting -1.
var s = 'just a test string';
console.log(s.search('/test/'));//-1
console.log(s.search(/test/));//7
Of course, the string search method can work with a string, in which case it will search for that specific substring, which in your case does not exist, so it returns the -1 index. In your example slashes were being added to the string, rather than producing the intended regular expression.
In JavaScript, there are two ways of creating a regular expression object (short of using code evaluation), a regular expression literal, and one created by the RegExp constructor.
A regular expression literal has to be defined at compile time, and cannot be constructed from string concatenation.
/test/i
To dynamically create a regular expression at runtime, you have to use the RegExp constructor.
RegExp('test', 'i');