smc

Let the roots of the quadratic be $p$ and $q$. Then, by Vieta's Formulas, $b=-(p+q)$ and $c=pq$. By substituting these values of $b$ and $c$ into our expression for $s$, we see that $s=pq-p-q+1$. By SFFT, $s=(p-1)(q-1)$. From the problem, we know that $p$ and $q$ are both greater than $1$, so $(p-1)$ and $(q-1)$ are necessarily positive. Thus, $s$, the product of these two positive terms, $\boxed{\text{ must be greater than zero}}$.