  Vector-boson scattering into two Z bosons at the LHC is a key channel for the
exploration of the electroweak sector of the Standard Model. It allows for the
full reconstruction of the scattering process but at the price of a huge
irreducible background. For the first time, we present full
next-to-leading-order predictions for $pp \to e^+e^-\mu^+\mu^-jj+X$ including
all electroweak and QCD contributions for vector-boson scattering signal and
irreducible background. The results are presented in the form of cross sections
and differential distributions. A particular emphasis is put on the newly
computed $O(\alpha_s^2 \alpha^5)$ corrections.
