The right quartet inside prosper (6)
I believe the answer is:
thrive
'prosper' is the definition.
('thrive' can be a synonym of 'prosper')
'the right quartet inside' is the wordplay.
'right' becomes 'r' (common abbreviation).
'quartet' becomes 'iv' (roman numerals for four).
'inside' indicates putting letters inside.
'r'+'iv'='riv'
'the' going around 'riv' is 'THRIVE'.
(Other definitions for thrive that I've seen before include "Prosper, grow well" , "Grow strongly" , "Grow vigorously, flourish" , "Do well, flourish" , "Boom" .)