How do i concatenate these two grep regexes

By.. concatenating the patterns?

grep -e '[Yy].*[Yy].[Ee][Ee][Ee][Ee].*[Yy].*[Yy]' first.txt

Or did you mean essentially doing a logical AND of the two patterns?

If the latter, you need to fake it, as while grep has built-in OR (|) and NOT (-v; [^]), it does not have a built-in AND. One way is by piping the output of one grep into the other:

grep -e '[Yy].*[Yy].[Ee][Ee]' first.txt | grep '[Ee][Ee].*[Yy].*[Yy]' 

The other way is to look for both patterns in series, in either order, with a logical OR (abbreviated for brevity):

grep -Ee 'pattern1.*pattern2|pattern2.*pattern1' input.txt

I find the first to be more succinct and easier to maintain.