We report the first observation of B-0 -> X(3872)(K+pi(-)) and evidence for B+ -> X(3872)(K-0 pi(+)). We measure the product of branching fractions for the former to be B(B-0 -> X(3872)(K+pi(-)) x B(X(3872) -> J/psi pi(+)pi(-)) = (7.9 +/- 1.3(stat) +/- 0.4(syst) x 10(-6) and find that B-0 -> X(3872)K*(892)(0) does not dominate the B-0 -> X(3872)K+pi(-) decay mode. We also measure B(B+ -> X(3872)(K-0 pi(+))) x B(X(3872) -> J/psi pi(+)pi(-)) = (10.6 +/- 3.0(stat) +/- 0.9(syst) x 10(-6). This study is based on the full data sample of 711 fb(-1) (772 x 10(6)B (B) over bar pairs) collected at the Upsilon(4S) resonance with the Belle detector at the KEKB collider.