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

 Routh array is given by

$\left.\begin{array}{ cccccc} s^5 \\ s^4 \\ s^3 \\ s^2 \\ s \\ s^0 \end{array}\right\vert{}\begin{array}{ cccccc} 1 & 3 & 4 \\ 4 & 1 & 1 \\ 2.75 & 3.75 & \\ -4.45 & 1 & \\ 4.367 & & \\ 1 & & \end{array}$

∴ There is sign change in the 1st column of routh array

∴ The polynomial is not Hurtwitz

(2) $p(s)= s^4 + 4s^3 +8$

The given polynomial contain even function only

$\therefore{}p^{'\left(s\right)}=s^4+4s^2+8 $

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

∴ There is sign change in the 1st column of routh array

∴ The polynomial is not Hurwitz