Distinct sets of elements are produced from different nucleosynthesis processes in galaxies that occur in core-collapse supernovae (CCSNe), Type-Ia supernovae (SNe Ia), asymptotic-giant-branch stars (AGBs), and various other enrichment sources. I will discuss both supernova and AGB enrichment in galaxies, as probed from integrated deep emission-line spectra of star-forming galaxies (SFGs) with direct elemental abundances. Galactic chemical enrichment from SNe has historically been constrained by alpha-enrichment ([α/Fe]) and metallicity ([Fe/H]) measurements from deep absorption-line spectra of individual stars in the Milky Way (MW) and some local group dwarf galaxies (and a handful of massive ellipticals with deep integrated absorption-line spectra out to z~2). The vast majority of galaxies in the universe are SFGs, with their fraction increasing with increasing redshift. We have recently shown that for SFGs (having deep integrated spectra with temperature sensitive auroral lines, enabling direct abundance determination), the oxygen-to-argon abundance ratio, log(O/Ar), vs Ar abundance, 12+log(Ar/H), is analogous to [α/Fe] vs [Fe/H] for stars. At low-z (z<0.3) with SDSS observations of ~800 SFGs, we show that galaxy chemical enrichment history is driven primarily by the interplay of CCSNe and SNe Ia, with their impact varying with galaxy mass. With a smaller sample of 11 SFGs at high-z (z~1.3-7.7) with JWST/NIRSPEC and Keck/MOSFIRE, we show that MW-like CCSNe and SNe Ia dominated enrichment processes occur at least out to z~4, beyond which rapid but intermittent star-formation may be at play. On the other hand, AGB nucleosynthesis is probed in SFGs from relative abundances of N & O. NIRSPEC@JWST observations revealed a handful of SFGs at high-z with high N/O abundance ratio at low O/H, dubbed extreme N-emitters. Some attribute this to extreme enrichment mechanisms active only in the early universe. However, we found high N/O at low O/H for a sample of 19 low-z (z<0.5) SFGs (a five-fold increase compared to earlier) using DESI DR1 spectra. The enhanced N/O values can be explained using galactic chemical evolution (GCE) models having long-lived N-enhancement from AGBs, coupled with strong outflows.