A quick sign check is useful, but stopping there can approve an unstable design. Routh makes the missing condition visible before a time simulation hides it behind a short recording.
Cross the boundary with one coefficient
Set a=2, b=3, c=4 in the Routh lab. Predict that the right-half-plane root count will increase, record the baseline, and move c to 8. The first column changes from [1,2,1,4] to [1,2,−1,8]. Two sign changes reveal two unstable roots even though the original polynomial coefficients remain positive.
Zero unstable roots does not mean strict stability
At c=6 the polynomial factors as (s+2)(s²+3). There are no strictly right-half-plane roots, but the pair ±j√3 lies on the boundary. The zero row and boundary message matter as much as the root count. At c=0 an origin pole gives another non-asymptotic case. Do not convert either case into a “stable” checkbox.
After the stability check
For this positive-a/b family, strict stability requires 0<c<ab. Then inspect how close poles are to the imaginary axis and whether the response meets settling and effort requirements. Routh counts unstable roots; it does not measure gain margin, phase margin or robustness to an uncertain physical parameter.
Inspect boundary modes as well as sign changes.