  We consider a nonlinear dispersive equation with a quasilinear quadratic
term. We establish two results. First, we show that solutions to this equation
with initial data of order $\mathcal{O}(\varepsilon)$ in Sobolev norms exist
for a time span of order $\mathcal{O}(\varepsilon^{-2})$ for sufficiently small
$\varepsilon$. Secondly, we derive the Nonlinear Schr\"odinger (NLS) equation
as a formal approximation equation describing slow spatial and temporal
modulations of the envelope of an underlying carrier wave, and justify this
approximation with the help of error estimates in Sobolev norms between exact
solutions of the quasilinear equation and the formal approximation obtained via
the NLS equation. The proofs of both results rely on estimates of appropriate
energies whose constructions are inspired by the method of normal-form
transforms. To justify the NLS approximation, we have to overcome additional
difficulties caused by the occurrence of resonances. We expect that the method
developed in the present paper will also allow to prove the validity of the NLS
approximation for a larger class of quasilinear dispersive systems with
resonances.
