  We present an analysis of a newly discovered 2+1+1 quadruple system with TESS
containing an unresolved eclipsing binary (EB) as part of TIC 121088960 and a
close neighbor TIC 121088959. The EB consists of two very low-mass M dwarfs in
a highly-eccentric ($e$ = 0.709) short-period ($P$ = 3.04358 d) orbit. Given
the large pixel size of TESS and the small separation (3.9$"$) between TIC
121088959 and TIC 121088960, we used light centroid analysis of the difference
image between in-eclipse and out-of-eclipse data to show that the EB likely
resides in TIC 121088960, but contributes only $\sim$10% of its light. Radial
velocity data were acquired with iSHELL at NASA's Infrared Facility and the
Coud${\'e}$ spectrograph at the McDonald 2.7-m telescope. For both images, the
measured RVs showed no variation over the 11-day observational baseline, and
the RV difference between the two images was $8 \pm 0.3$ km s$^{-1}$. The
similar distances and proper motions of the two images indicate that TIC
121088959 and TIC 121088960 are a gravitationally bound pair. Gaia's large RUWE
and astrometric_excess_noise parameters for TIC 121088960, further indicate
that this image is the likely host of the unresolved EB and is itself a triple
star. We carried out an SED analysis and calculated stellar masses for the four
stars, all of which are in the M dwarf regime: 0.19 M$_\odot$ and 0.14
M$_\odot$ for the EB stars and 0.43 M$_\odot$ and 0.39 M$_\odot$ for the
brighter visible stars, respectively. Lastly, numerical simulations show that
the orbital period of the inner triple is likely the range 1 to 50 years.
