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

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

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

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

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

$f(-1)=0$

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

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

$p+q=-10$

$\underline{-p+q=-8}$

$2q=-18$

$q=-9$

$p=-1$