We present a coherent analysis of Higgs-boson production in the channels $\mathrm{pp}\ensuremath{\rightarrow}\mathrm{ZZX}\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}}{l}^{\ensuremath{'}+}{l}^{\ensuremath{'}\ensuremath{-}}X(l, {l}^{\ensuremath{'}}=e, \ensuremath{\mu})$ and $\mathrm{pp}\ensuremath{\rightarrow}\mathrm{ZZX}\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}}\ensuremath{\nu}\overline{\ensuremath{\nu}}X(\ensuremath{\nu}={\ensuremath{\nu}}_{e}, {\ensuremath{\nu}}_{\ensuremath{\mu}}, {\ensuremath{\nu}}_{\ensuremath{\tau}})$ for ${m}_{H}\ensuremath{\ge}600$ GeV at hadron supercolliders, using the exact matrix elements for $\mathrm{gg}\ensuremath{\rightarrow}\mathrm{ZZ}$ and $\mathrm{qq}\ensuremath{\rightarrow}\mathrm{qqZZ}$. The importance of a complete understanding of the shape of the perturbative $\mathrm{pp}\ensuremath{\rightarrow}\mathrm{ZZX}$ background from nonresonant diagrams is emphasized. We find that for the CERN Large Hadron Collider (LHC) to have a Higgs-boson discovery potential comparable to that of the Superconducting Super Collider (SSC) requires at least a factor of 10 times more integrated luminosity. In particular, assuming an integrated luminosity of ${10}^{4}$ ${\mathrm{pb}}^{\ensuremath{-}1}$ and perfect lepton identification efficiency, the LHC (SSC) can identify Higgs bosons as resonances with mass up to 600 (800) GeV in $\mathrm{pp}\ensuremath{\rightarrow}\mathrm{ZZX}\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}}{l}^{\ensuremath{'}+}{l}^{\ensuremath{'}\ensuremath{-}}X$. To extend the discovery range of the LHC to ${m}_{H}=800$ GeV in this channel requires an integrated luminosity of at least ${10}^{5}$ ${\mathrm{pb}}^{\ensuremath{-}1}$. For ${m}_{H}>800$ GeV, a clear resonance structure is missing; however, one can still discriminate between a heavy Higgs boson with ${m}_{H}\ensuremath{\sim}1$ TeV and a light Higgs boson (${m}_{H}\ensuremath{\lesssim}2{M}_{z}$) at the SSC.