  The Carina dwarf spheroidal (dSph) galaxy is the only galaxy of this type
that shows clearly episodic star formation separated by long pauses. Here we
present metallicities for 437 radial velocity members of this Galactic
satellite. We obtained medium-resolution spectroscopy with the multi-object
spectrograph FLAMES at the ESO VLT. Our target red giants cover the entire
projected surface area of Carina. Our spectra are centered at the near-infrared
Ca triplet, which is a well-established metallicity indicator for old and
intermediate-age red giants. The resulting data sample provides the largest
collection of spectroscopically derived metallicities for a Local Group dSph to
date. Four of our likely radial velocity members of Carina lie outside of this
galaxy's nominal tidal radius, supporting earlier claims of the possible
existence of such stars beyond the main body of Carina. We find a mean
metallicity of [Fe/H]=-1.7 dex. The formal full width at half maximum of the
metallicity distribution is 0.92 dex, while the full range of metallicities
spans ~-3.0<[Fe/H]<0.0 dex. The metallicity distribution might be indicative of
several subpopulations. There appears to be a mild radial gradient such that
more metal-rich populations are more centrally concentrated, matching a similar
trend for an increasing fraction of intermediate-age stars. This as well as the
photometric colors of the more metal-rich red giants suggest that Carina
exhibits an age-metallicity relation. Indeed the age-metallicity degeneracy
seems to conspire to form a narrow red giant branch despite the considerable
spread in metallicity and wide range of ages. The metallicity distribution is
not well-matched by a simple closed-box model of chemical evolution, but
requires models that take into account also infall and outflows. (Abridged)

Astrophysics