Routh stability criterion

S^5+ 4S^4+2S^3 +S^2+2S+ k=0 range of K to be stable

The matrices for your equation will be:

1 2 2

4 1 k

7/4 (8-k)/4 0

(4k-25)/7


According to Routh–Hurwitz stability criterion, the signs of the first column parameters must be the same. So:

(4k-25)/7 must be greater than zero >>>>> 4k-25>0 >>>>> k>6.25

K must be also positive since all the coefficients of the original equation must be the same sign.

So: 0