The condition that the product of two of the roots $x^3+p{x}^2+qx+r=0$  is $-1$ , is

Roots are $\alpha ,\beta ,\gamma$

Given $\alpha \beta =-1$

$\alpha \beta \gamma =-r$

$\left(-1\right)\gamma =-r$

$\gamma =r$

Substitute in equation

$f\left(\gamma \right)=r^3+p{r}^2+qr+r=0$

$f\left(\gamma \right)=r^2+pr+q+1=0$