POLYnomials

Since the roots are in geometric progression, we can write them as $\{ tu^{-2},tu^{-1},t,tu,tu^2 \}$ . Note that none of the roots can be zero (otherwise the reciprocals wouldn't be defined), so this representation is valid. The reason for this choice is it makes the algebra a little easier:

The sum of the roots is $t \left(u^{-2}+u^{-1}+1+u+u^2 \right) = 40$ (using Vieta).

The sum of the reciprocals of the roots is $\frac{1}{t} \left(u^{2}+u+1+u^{-1}+u^{-2} \right) = 10$ (given in the problem).

From these, we get $t^2=4$ , so that $t=\pm 2$ .

The product of the roots (using Vieta again) is just $t^5=-s$ ; from above, this means that $|s|=2^5=\boxed{32}$ .