Charm meson oscillations are observed in a time-dependent analysis of the ratio of $D^0to K^+pi^-pi^+pi^-$ to $D^0to K^-pi^+pi^-pi^+$ decay rates, using data corresponding to an integrated luminosity of $3.0,{rm fb}^{-1}$ recorded by the LHCb experiment. The measurements presented are sensitive to the phase-space averaged ratio of doubly Cabibbo-suppressed to Cabibbo-favoured amplitudes $r_{D}^{K3pi}$ and the product of the coherence factor $R_{D}^{K3pi}$ and a charm mixing parameter $y^{}_{K3pi}$. The constraints measured are $r_{D}^{K3pi}=(5.67 pm 0.12)times10^{-2}$, which is the most precise determination to date, and $R_{D}^{K3pi} cdot y^{}_{K3pi} = (0.3 pm 1.8)times 10^{-3}$, which provides useful input for determinations of the CP-violating phase $gamma$ in $B^pm to D K^pm, D to K^mppi^pmpi^mppi^pm$ decays. The analysis also gives the most precise measurement of the $D^0to K^+pi^-pi^+pi^-$ branching fraction, and the first observation of $D^0-bar D^0$ oscillations in this decay mode, with a significance of 8.2 standard deviations.
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