Routh's stability criterion: zeros of the auxiliary polynomial and of its derivative
The zeros of the auxiliary polynomial give you the poles of equal magnitude and opposite signs (either on the real axis or on the imaginary axis). The derivative of the auxiliary polynomial gives you the coefficients that are used in the row with all zero coefficients. It's been a while, but I learned that from reading Modern Control Engineering by K. Ogata (and I had to refresh my memory for answering this question).