This note considers the stability of linear systems with a time-varying delay. We are interested in a simple Lyapunov-Krasovskii functional (LKF) approach without delay decomposition. In this category, all recent tractable results had a fixed bound on the allowable maximum size of the delay for years. We propose a new simple LKF including the cross terms of variables and quadratic terms multiplied by a higher degree scalar function, and present a new result expressed in the form of LMIs. We show, by two well-known examples, that our result overcomes the previous allowable maximum size of delay and it is less conservative than the previous results having a relatively small upper bound in the derivative of time-delay. (C) 2011 Elsevier Ltd. All rights reserved.