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Using 2.93 fb−1 of 𝑒+𝑒− collision data collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, the first observation of the doubly Cabibbo-suppressed decay 𝐷+→𝐾+𝜋+𝜋−𝜋0 is reported. After removing decays that contain narrow intermediate resonances, including 𝐷+→𝐾+𝜂, 𝐷+→𝐾+𝜔, and 𝐷+→𝐾+𝜙, the branching fraction of the decay 𝐷+→𝐾+𝜋+𝜋−𝜋0 is measured to be (1.13±0.08stat±0.03syst)×10−3. The ratio of branching fractions of 𝐷+→𝐾+𝜋+𝜋−𝜋0 over 𝐷+→𝐾−𝜋+𝜋+𝜋0 is found to be (1.81±0.15)%, which corresponds to (6.28±0.52)tan4𝜃𝐶, where 𝜃𝐶 is the Cabibbo mixing angle. This ratio is significantly larger than the corresponding ratios for other doubly Cabibbo-suppressed decays. The asymmetry of the branching fractions of charge-conjugated decays 𝐷±→𝐾±𝜋±𝜋∓𝜋0 is also determined, and no evidence for 𝐶𝑃 violation is found. In addition, the first evidence for the 𝐷+→𝐾+𝜔 decay, with a statistical significance of 3.3𝜎, is presented and the branching fraction is measured to be ℬ(𝐷+→𝐾+𝜔) = (5.7+2.5−2.1stat±0.2syst)×10−5.