Quintic root composition

Since $r_1$ , $r_2$ , $r_3$ , $r_4$ , and $r_5$ are the roots of $P(x)$ , this means that

$\begin{aligned} \prod_{k=1}^5 (x-r_k) & = P(x) = x^5 - x^2 + 1 \\ \prod_{k=1}^5 (r_k+x) & = - P(x) \\ \prod_{k=1}^5 (r_k+i) & = - P(i) \\ \prod_{k=1}^5 (r_k+i)(r_k-i) & = P(i)P(-i) \\ \prod_{k=1}^5 (r_k^2+1) & = (i^5-i^2+1)((-i)^5-(-i)^2+1) \\ \prod_{k=1}^5 Q(r_k) & = (i-(-1)+1)(-i-(-1)+1) \\ & =(2+i)(2-i) = \boxed 5 \end{aligned}$