Problem in area backing hostility (6)
I believe the answer is:
animus
'hostility' is the definition.
(animus is a kind of hostility)
'problem in area backing' is the wordplay.
'problem' becomes 'sum' (arithmetic problem).
'area' becomes 'a' (maths abbreviation).
'backing' says the letters should be written in reverse.
'sum'+'in'+'a'='sumina'
'sumina' backwards is 'ANIMUS'.
(Other definitions for animus that I've seen before include "Hostile spirit in speech or action" , "Heat" , "Hostility shown in speech or action" , "Ill-feeling; hatred" , "Motivating spirit" .)