We report an analysis of charmless hadronic decays of charged B mesons to the final state K + π 0 π 0 , using a data sample of ( 470.9 ± 2.8 ) × 10 6 B ¯¯¯ B events collected with the BABAR detector at the Υ ( 4 S ) resonance. We observe an excess of signal events, with a significance above 10 standard deviations including systematic uncertainties, and measure the branching fraction and C P asymmetry to be B ( B + → K + π 0 π 0 ) = ( 16.2 ± 1.2 ± 1.5 ) × 10 − 6 and A C P ( B + → K + π 0 π 0 ) = − 0.06 ± 0.06 ± 0.04 , where the uncertainties are statistical and systematic, respectively. Additionally, we study the contributions of the B + → K ∗ ( 892 ) + π 0 , B + → f 0 ( 980 ) K + , and B + → χ c 0 K + quasi-two-body decays. We report the world's best measurements of the branching fraction and C P asymmetry of the B + → K + π 0 π 0 and B + → K ∗ ( 892 ) + π 0 channels.