  Of particular interest for radio and hard X-ray diagnostics of accelerated
electrons during solar flares is the understanding of the basic non-linear
mechanisms regulating the relaxation of electron beams propagating in turbulent
plasmas. In this work, it is shown that in addition to scattering of beam
electrons, scattering of the beam-generated Langmuir waves via for instance
mode-coupling, can also result in broadening of the wave-particle resonance. We
obtain a resonance-broadened version of weak-turbulence theory with
mode-coupling to ion-sound modes. Resonance broadening is presented here as a
unified framework which can quantitatively account for the reduction and
possible suppression of the beam instability due to background scattering of
the beam electrons themselves or due to scattering of the beam-generated
Langmuir waves in fluctuating plasmas. Resonance broadening being essentially
equivalent to smoothing of the electron phase-space distribution, it is used to
construct an intuitive physical picture for the stability of inverted
populations of fast electrons that are commonly observed \emph{in-situ} to
propagate in the solar-wind.
