This article presents a new framework for dealing with pure and distributed delays simultaneously and impartially as general delays in algebraic finite spectrum assignment. The framework based on an important property common to general delays gives a sufficient condition for finite spectrum assignability of scalar systems with several general delays and a design method of a controller achieving finite spectrum assignment. Especially in algebraic control theory for delay systems, pure and distributed delays have never been dealt with simultaneously and impartially. The framework is significantly important also for multidimensional system theory because its potential applicability to the discussion on systems with general delays, i.e. the possibility that especially distributed delay operators are used as variables for describing multidimensional systems, is clarified for the first time.