  We report on a recent calculation of next-to-leading-order (NLO) QCD and
electroweak corrections to like-sign W-boson scattering at the Large Hadron
Collider, including all partonic channels and W-boson decays in the process $pp
\to e^+ \nu_e \mu^+ \nu_\mu jj + X$. The calculation is implemented in the
Monte Carlo integrator Bonsay and comprises the full tower of NLO contributions
of the orders $\alpha_s^3\alpha^4$, $\alpha_s^2\alpha^5$, $\alpha_s\alpha^6$,
and $\alpha^7$. Our numerical results confirm and extend previous results, in
particular the occurrence of large purely electroweak corrections of the order
of $\sim-12\%$ for integrated cross sections, which get even larger in
distributions. We construct a "VBS approximation" for the NLO prediction based
on partonic channels and gauge-invariant (sub)matrix elements potentially
containing the vector-boson scattering (VBS) subprocess and on resonance
expansions of the Wdecays. The VBS approximation reproduces the full NLO
predictions within $\sim1.5\%$ in the most important regions of phase space.
Moreover, we discuss results from different versions of "effective vector-boson
approximations" at leading order, based on the collinear emission of W bosons
of incoming (anti)quarks. However, owing to the only mild collinear enhancement
and the design of VBS analysis cuts, the quality of this approximation turns
out to be only qualitative at the LHC.
