How do i find the greatest common factor of 14m², 27a³, and 45a²?

Exactly the way you do it with numbers:

-- List all the factors of the first one.
-- List all the factors of the second one.
-- List all the factors of the third one.

-- Make a short list, consisting of the factors that appear
   on all three lists.  Those are the 'common' factors.

-- The greatest item on the short list is the greatest common factor. 

14m²,  27a³, and  45a²  don't have any common factor except ' 1 '.
Since ' 1 ' is their only common factor, it's also their greatest one.