(i)  $p\left(s\right)=s^4+4s^2+8$

The given polynomial contain even function only

$\therefore{}p^{'}\left(s\right)=4s^3+8s$

Routh array :

$\left.\begin{array}{ ccccc} s^4 \\ s^3 \\ s^2 \\ s \\ s^0 \end{array}\right\vert{}\begin{array}{ ccccc} 1 & 4 & 8 \\ 4 & 8 & \\ 2 & 8 & \\ -8 & & \\ 8 & & \end{array}$

$\because{}there\ is\ sign\ change\ in\ the\ 1^{st}column\ of\ routh\ array\ $

$\therefore{}polynomial\ is\ not\ Hurtwitz $

(ii)  $p\left(s\right)=s^4+s^3+5s^2+3s+4$

Routh array :

$\left.\begin{array}{ ccccc} s^4 \\ s^3 \\ s^2 \\ s \\ s^0 \end{array}\right\vert{}\begin{array}{ ccccc} 1 & 5 & 4 \\ 1 & 3 & \ \\ 2 & 4 & \ \\ 1 & \ & \ \\ 4 & \ & \ \end{array}$

∴ All the elements in 1st column are positive

$\therefore{}the\ polynominal\ is\ Hurtwitz $