A 2+1+1 quadruple star system containing the most eccentric, low-mass, short-period, eclipsing binary known.

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.