<p>A new necessary and sufficient stability test in a tractable number of operations for linear neutral-type delay systems is introduced. It is developed in the Lyapunov–Krasovskii framework via functionals with prescribed time derivative and the polynomial approximation theory. The substitution of any polynomial approximation of the functional argument derives a quadratic form, whose inner matrix is characterized by integrals of the delay Lyapunov matrix multiplied by monomials, independent of the coefficients of the approximation under consideration. In the particular case of Chebyshev polynomials as a basis for the polynomial approximation, a bound for the functional approximation error is determined and estimated on a special set of functions, delivering a positive semi-definiteness stability test in a finite number of mathematical operations. Some examples illustrate the obtained results.</p>