The authors present results from an analysis of B{sup 0} {yields} {rho}{sup +}{rho}{sup -} decays using (383.6 {+-} 4.2) x 10{sup 6} B{bar B} pairs collected by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. The measurements of the B{sup 0} {yields} {rho}{sup +}{rho}{sup -} branching fraction, longitudinal polarization fraction f{sub L}, and the CP-violating parameters S{sub long} and C{sub long} are: {Beta}(B{sup 0} {yields} {rho}{sup +}{rho}{sup -}) = (25.5 {+-} 2.1(stat){sub -3.9}{sup +3.6}(syst)) x 10{sup -6}, f{sub L} = 0.992 {+-} 0.024(stat){sub -0.013}{sup +0.026}(syst), S{sub long} = -0.17 {+-} 0.20(stat){sub -0.06}{sup +0.05}(syst), C{sub long} = 0.01 {+-} 0.15(stat) {+-} 0.06(syst). The authors determine the unitarity triangle angle {alpha}, using an isospin analysis of B {yields} {rho}{rho} decays. One of the two solutions, {alpha} = [73.1, 117.0]{sup o} at 68% CL is compatible with standard model-based fits of existing data. Constraints on the unitarity triangle are also evaluated using an SU(3) symmetry based approach.