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We present new integrated light spectroscopy of globular clusters (GCs) in NGC 5128, a nearby giant elliptical galaxy less than 4 Mpc away, in order to measure radial velocities and derive ages, metallicities, and alpha-element abundance ratios. Using the Gemini South 8 meter telescope with the instrument Gemini Multi-Object Spectrograph, we obtained spectroscopy in the range of ∼3400-5700 Å for 72 GCs with a signal-to-noise ratio greater than 30 Å^–1; and we have also discovered 35 new GCs within NGC 5128 from our radial velocity measurements. We measured and compared the Lick indices from Hδ_A through Fe5406 with the single stellar population models of Thomas et al. in order to derive age, metallicity, and [α/Fe] values. We also measure Lick indices for 41 Milky Way GCs from Puzia et al. and Schiavon et al. with the same methodology for direct comparison. Our results show that 68% of the NGC 5128 GCs have old ages (>8 Gyr), 14% have intermediate ages (5-8 Gyr), and 18% have young ages (<5 Gyr). However, when we look at the metallicity of the GCs as a function of age, we find 92% of metal-poor GCs and 56% of metal-rich GCs in NGC 5128 have ages >8 Gyr, indicating that the majority of both metallicity subpopulations of GCs formed earlier, with a significant population of young and metal-rich GCs forming later. Our metallicity distribution function generated directly from spectroscopic Lick indices is clearly bimodal, as is the color distribution of the same set of GCs. Thus, the metallicity bimodality is real and not an artifact of the color to metallicity conversion. However, the metallicity distribution function obtained from comparison with the single stellar population models is consistent with a unimodal, bimodal, or multimodal shape. The [α/Fe] values are supersolar with a mean value of 0.14±0.04, indicating a fast formation timescale. However, the GCs in NGC 5128 are not as [α/Fe] enhanced as the Milky Way GCs also examined in this study. Our measured indices also indicate that the GCs in NGC 5128 may have a slight overabundance in nitrogen and a wider range of calcium strength compared to the Milky Way GCs. Our results support a rapid, early formation of the GC system in NGC 5128, with subsequent major accretion and/or GC and star-forming events in more recent times.