This paper proposes a systematic method to analyse the stability of systems with single delay with coefficient polynomial depending on the delay. Such models often arise the describing dynamics in Life Science and Engineering systems. A method was presented by Beretta and Kuang in 2002. Their work extends their results to the general case with the exception of some degenerate cases. The interval of interest for the delay is partitioned to smaller intervals so that the magnitude condition generate a fixed number of frequencies as functions of the delay within each interval. The crossing conditions are expressed in a general form, and a simplified derivation for the first order derivative criterion is obtained.