  We study CP violation and the contribution of the strong pion-pion
interactions in the three-body B+- --> pi+- pi-+ pi+- decays within a quasi
two-body QCD factorization approach. The short distance interaction amplitude
is calculated in the next-to-leading order in the strong coupling constant with
vertex and penguin corrections. The meson-meson final state interactions are
described by pion non-strange scalar and vector form factors for the S and P
waves and by a relativistic Breit-Wigner formula for the D wave. The pion
scalar form factor is calculated from a unitary relativistic coupled-channel
model including pi pi, Kbar K and effective (2pi)(2pi) interactions. The pion
vector form factor results from a Belle Collaboration analysis of tau- --> pi-
pi0 nu_tau data. The recent B+- --> pi+- pi-+ pi+- BABAR Collaboration data are
fitted with our model using only three parameters for the S wave, one for the P
wave and one for the D wave. We find not only a sizable contribution of the S
wave just above the pi pi threshold but also under the rho(770) peak a
significant interference, mainly between the S and P waves. For the B to
f_2(1270) transition form factor, we predict F^{Bf_2}(m_\pi^2)=0.098+-0.007.
Our model yields a unified unitary description of the contribution of the three
scalar resonances f_0(600), f_0(980) and f_0(1400) in terms of the pion
non-strange scalar form factor.
