If $P(x)=0$ then $Q(1/x)=0$, there $Q(x)=P(1/x)\cdot x^{45}$.

Since the coefficients are positive, all the original roots are negative. Now we can separate the polynomials into pairs of $P(x)$ and $Q(x)$ where the coefficients of $Q(x)$ are the coefficients of $P(x)$ backwards. Then $P(x) = Q(1/x)\cdot x^{45}$, so for any root $r$ of $P(x)$, $1/r$ is a root of $Q(x)$. Therefore, for any root $r$ with $|r| > 1$, we can find a root $s$ with $|s| < 1$ and vice versa, so there are an equal number of positive and negative numbers.