The Galois group of the splitting field of $h$ over $\mathbf{Q}$ is $\mathbf{Z}/2 \times \mathbf{Z}/2$, the Klein four group. Since it is abelian, the splitting of primes is determined by congruence conditions, and the density of the splitting types is given by Chebotarev.

The Galois group of the splitting field of $h$ over $\mathbf{Q}$ is $\mathbf{Z}/2 \times \mathbf{Z}/2$, the Klein four group. Since it is abelian, the splitting of primes is determined by congruence conditions, and the density of the splitting types is given by Chebotarev. You can realise the splitting field as a subfield of $\mathbf{Q}(\zeta_{\mathrm{|disc|}})$ and use what you know about splitting of primes in cyclotomic extensions.