Why is it that $x^4+2x^2+1$ is reducible in $\mathbb{Z}[x]$ but has no roots in $\mathbb{Q}$?

There is no contradiction here. The polynomial $f(x)=x^4+2x^2+1$ is reducible both in $\mathbb{Q}[x]$ and in $\mathbb{Z}[x]$, since it can be factored as $(x^2+1)(x^2+1)$ in either ring. All that's going on is that a polynomial can be reducible without having any roots.

It is not necessary that a polynomial is reducible over a field iff it has a root in field. By meaning of a reducible polynomial is that we express the polynomial into product of non constant polynomials over the field. And a result that a polynomial having degree 2 or 3 is reducible iff it has root in field. In your example degree of polynomial is 4.