The system of delay differential equations which has its delay in highest derivative are known as neutral delay differential equations (NDDEs) which is found in many applied sciences branches which plays a crucial part in the modelling of real-world occurrences mathematically. Some methods were complicated to find an analytical solution approximately. In this study, using the generalized Lambert W function to construct the characteristic roots of the neutral delay differential equation with random delay of first order type and examples were derived to support the outcome numerically.