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Using $2.93 \rm fb^{-1}$ of $e^+e^-$ collision data collected at acenter-of-mass energy of 3.773 GeV with the BESIII detector, the firstobservation of the doubly Cabibbo-suppressed decay $D^+\to K^+\pi^+\pi^-\pi^0$is reported. After removing decays that contain narrow intermediate resonances,including $D^+\to K^+\eta$, $D^+\to K^+\omega$, and $D^+\to K^+\phi$, thebranching fraction of the decay $D^+\to K^+\pi^+\pi^-\pi^0$ is measured to be$(1.13 \pm 0.08_{\rm stat} \pm 0.03_{\rm syst})\times 10^{-3}$. The ratio ofbranching fractions of $D^+\to K^+\pi^+\pi^-\pi^0$ over $D^+\toK^-\pi^+\pi^+\pi^0$ is found to be $(1.81\pm0.15)$\%, which corresponds to$(6.28\pm0.52)\tan^4\theta_C$, where $\theta_C$ is the Cabibbo mixing angle.This ratio is significantly larger than the corresponding ratios for otherdoubly Cabibbo-suppressed decays. The asymmetry of the branching fractions ofcharge-conjugated decays $D^\pm\to K^\pm\pi^\pm\pi^\mp\pi^0$ is alsodetermined, and no evidence of $CP$ violation is found. In addition, the firstevidence of the $D^+\to K^+\omega$ decay, with a statistical significance of3.3$\sigma$, is presented and its decay branching fraction is determined to be$({5.7^{+2.5}_{-2.1}}_{\rm stat}\pm0.2_{\rm syst})\times10^{-5}$.

Observation of the Doubly Cabibbo-Suppressed Decay $D^+\to K^+\pi^+\pi^-\pi^0$ and Evidence for $D^+\to K^+\omega$