The roots of a polynomial always come in complex conjugate pairs. Therefore there are zero, two or four real roots. Assume the general case that the roots are a+-bi and c+-di. The product of all the roots must equal -1. Therefore -1 = a^2 + b^2 +c^2 +d^2. All real numbers have a square>=0, therefore at least one of the terms a,b,c or d must be imaginary, and therefore at least 2 must be imaginary. Therefore there can be a maximum of only 2 real roots.
Solution (spoiler) - I think
