Solution:

i)

$Q(S)=S^5+s^3+s^1$

$Q^1(s)=5 s^4+3 s^2+1$

Using Routh Array method

$ \begin{array}{c|ccc} s^5 & 1 & 1 & 1 \\\\ \hline S^4 & 5 & 3 & 1 \\\\ \hline S^3 & 2 / 5 & 4 / 5 & \\\\ \hline s^2 & -7 & & \\\\ \hline s^1 & & \\\\ \hline s^0 & & \\\\ \end{array} \\ $

$\because$ first element in row of. $s^2$ is -ve. It is not Hurwitz's polynomial.

ii) $ Q(s)=s^4+6 s^3+8 s^2+10 $

$ \begin{aligned} \\ &m(s)=s^4+8 S^2 \\\\ &n(s)=6 S^3+10 \\ \end{aligned} \\ $

Using Routh array method

$ \begin{array}{c|cc} s^4 & 1 & 8 \\\\ \hline s^3 & 6 & 10 \\\\ \hline s^2 & 6.33 & \\\\ \hline s^1 & \\\\ \hline s^0 & & \\\\ \end{array} \\ $

In given $Q(s)$ $s^1$ term is missing. Hence It is not Hurwitz's polynomial.