Egghead at research facility to give speech that is quite complicated (9)
I believe the answer is:
elaborate
'is quite complicated' is the definition.
(I've seen this in another clue)
'egghead at research facility to give speech' is the wordplay.
'egghead' becomes 'e' (head letter of 'egg').
'at' means one lot of letters go next to another.
'research facility' becomes 'lab' (laboratory).
'to give speech' becomes 'orate' (I've seen this before).
'e'+'lab'+'orate'='ELABORATE'
'that' acts as a link.
(Other definitions for elaborate that I've seen before include "very busy" , "Complicated or fancy" , "Work out or describe in detail" , "Add details to clarify meaning" , "Ostentatious, possibly" .)