Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and $mu$, $ u$ be a pair of generalized derivations on R. If < $mu^2(x)+ u(x),;x^n$ > = 0 for all x $in$ R, then $mu$ and $ u$ are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center $C_R$ and d, g be a pair of derivations on R. If < $d^2(x)+g(x)$, $x^n$ > $in$ $C_R$ for all x $in$ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.