Using division algorithm, the values of $p$ and  $q$, such that  $x^4+x^3+8x^2+px+q$ is exactly divisible by  $x^2-1$ is 

$f\left(x\right)=x^4+x^3+8x^2+px+q$ is exactly divisible by  $x^2-1$ is

$g\left(x\right)=x^2-1$

$f\left(1\right)=0$

$1+1+8+p+q=0$

$p+q=-10\to \left(1\right)$

$f(-1)=0$

$1-1+8-p+q=0$

$-p+q=-8\to \left(2\right)$

$\begin{array}{l}p+q=-10\\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{-p+q=-8}\\ 2q=-18\end{array}$

$q=-9$

$p=-1$