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Abstract 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 → e+e − μ + μ − 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 ( α s 2 $$ {\alpha}_{\mathrm{s}}^2 $$ α 5) corrections.