Determined to find the answer in 'Scarlet' (8)
I believe the answer is:
resolved
'determined' is the definition.
(resolving is a kind of determining)
'the answer in scarlet' is the wordplay.
'the answer' becomes 'solve'.
'in' indicates putting letters inside.
'scarlet' becomes 'red' ('red' can be a synonym of 'scarlet').
'solve' placed within 'red' is 'RESOLVED'.
'to find' acts as a link.
(Other definitions for resolved that I've seen before include "Decided firmly on a course of action" , "Settled, found a solution" , "Determind - mind made up" , "Determined - found the answer again" , "Decided, determined" .)