In this paper, we consider the initial-boundary value problem of the following semilinear heat equation with past and finite history memories: \t\t\tut−Δu+∫0tg1(t−s)div(a1(x)∇u(s))ds+∫0+∞g2(s)div(a2(x)∇u(t−s))ds+f(u)=0,(x,t)∈Ω×[0,+∞),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} &u_{t}-\\Delta u + \\int _{0}^{t} {{g_{1}}(t - s) \\operatorname{div}\\bigl({a_{1}}(x) \\nabla u(s)\\bigr)\\,ds} \\\\ &\\quad{} + \\int _{0}^{ + \\infty } {{g_{2}}(s) \\operatorname{div}\\bigl({a_{2}}(x)\\nabla u(t - s)\\bigr)\\,ds}+ f(u)=0, \\quad(x,t)\\in \\varOmega \\times [0,+\\infty ), \\end{aligned}$$ \\end{document} where Ω is a bounded domain. Under suitable conditions on a_{1} and a_{2}, for a large class of relation functions g_{1} and g_{2}, we establish a general decay estimate, including the usual exponential and polynomial decay cases.