  We present a new calculation of next-to-leading-order corrections of the
strong and electroweak interactions to like-sign W-boson scattering at the
Large Hadron Collider, implemented in the Monte Carlo integrator Bonsay. The
calculation includes leptonic decays of the $\mathrm{W}$ bosons. It comprises
the whole tower of next-to-leading-order contributions to the cross section,
which scale like $\alpha_\mathrm{s}^3\alpha^4$, $\alpha_\mathrm{s}^2\alpha^5$,
$\alpha_\mathrm{s}\alpha^6$, and $\alpha^7$ in the strong and electroweak
couplings $\alpha_\mathrm{s}$ and $\alpha$. We present a detailed survey of
numerical results confirming the occurrence of large pure electroweak
corrections of the order of $\sim-12\%$ for integrated cross sections and even
larger corrections in high-energy tails of distributions. The electroweak
corrections account for the major part of the complete next-to-leading-order
correction, which amounts to $15{-}20\%$ in size, depending on the details of
the event selection chosen for analysing vector-boson-scattering. Moreover, we
compare the full next-to-leading-order corrections to approximate results based
on the neglect of contributions that are not enhanced by the vector-boson
scattering kinematics (VBS approximation) and on resonance expansions for the
$\mathrm{W}$-boson decays (double-pole approximation); the quality of this
approximation is good within $\sim 1.5\%$ for integrated cross sections and the
dominating parts of the differential distributions. Finally, for the
leading-order predictions, we construct different versions of effective
vector-boson approximations, which are based on cross-section contributions
that are enhanced by collinear emission of $\mathrm{W}$ bosons off the
initial-state (anti)quarks; in line with previous findings in the literature,
it turns out that the approximative quality is rather limited for applications
at the LHC.
