Showing one's work, on the way out without greeting head of Bauhaus (10)
I believe the answer is:
exhibiting
'showing one's work' is the definition.
'exhibiting' can be an answer for 'showing' (exhibiting is a kind of showing). I am unsure of the 'one's work' bit.
'on the way out without greeting head of bauhaus' is the wordplay.
'on the way out' becomes 'exiting' (I've seen this in another clue).
'without' indicates putting letters inside ('without' can be similar in meaning to 'outside').
'greeting' becomes 'hi' (hi is a kind of greeting).
'head of' indicates taking the first letters.
The initial letter of 'bauhaus' is 'b'.
'hi'+'b'='hib'
'exiting' enclosing 'hib' is 'EXHIBITING'.
Can you help me to learn more?
(Another definition for exhibiting that I've seen is " demonstrator's job".)