Delay differential system model can often be found in control process, physics,chemical engineering and ecology. It is more significant to study the properties of the system and establish easily testable algebraic criteria. At first, the delay-independent stability and the delay-dependent stability of a class of neutral delay differential systems are considered respectively, and the corresponding algebraic criteria and algorithms are presented in this paper. Afterwards, the Hopf bifurcation of a class of delay differential systems is studied, and the corresponding algebraic criteria and algorithms are presented. The main contributions can be summarized as follows. 1, The delay-independent stability of a class of neutral multi-delay differential systems is analyzed. By using the "Complete Discrimination System for Polynomials", easily testable algebraic criteria and algorithms for delay-independent stability of the systems are established after considering the global hyperbolicity of the related difference systems. The results generalize and unify the relevant methods in the literature. Especially, the results can be applied to the related class of retarded differential systems completely. Some numerical examples are provided to illustrate the effectiveness of the results. 2, Based on the work mentioned above, the delay-dependent stability of the class of neutral multi-delay differential systems is discussed. That is, the delay intervals for stability of the systems can be computed if the systems are not delay-independent stable tested by the algebraic criteria mentioned above. The corresponding algebraic criteria and algorithms are presented, and the results can be applied to the related class of retarded differential systems completely. A numerical example is provided to illustrate the results. 3, The general method, used in the most literature for determining the occurrence of Hopf bifurcation for delay differential systems, is summarized and generalized. The literature has just considered about Hopf bifurcation for the class of lower-dimensional one-delay differential systems. Especially, the paper summarized the method used in the literature, and turned the problem into determining the real roots of a one-variable polynomial equation. In this paper, the general method is summarized again for a class of higher-dimensional multi-delay differential systems by using a different transformation, and turn the bifurcation analysis into determining the real roots of a pair of two-variable polynomial equations.