This article studies the stability problem of generalized neural networks (GNNs) with time-varying delay. The delay has two cases: the first case is that the delay's derivative has only upper bound, the other case has no information of its derivative or itself is not differentiable. For both two cases, we provide novel stability criteria based on novel Lyapunov-Krasovskii functionals (LKFs) and new negative definite conditions (NDCs) of matrix-valued cubic polynomials. In contrast with the existing methods, in this article, the proposed criteria do not need to introduce extra state variables, and the positive-definite constraint on the novel LKF is relaxed. Moreover, based on free-matrix-based inequality (FMBI) and new NDCs, the stability conditions are expressed as linear matrix inequalities (LMIs). Eventually, the merits and efficiency of the proposed criteria are checked through some classical numerical examples.