The metallicity of active galactic nuclei (AGNs), which can be measured by emission line ratios in their broad- and narrow-line regions (BLRs and NLRs), provides invaluable information about the physical connection between the different components of AGNs. From the archival data bases of the International Ultraviolet Explorer, the Hubble Space Telescope and the Sloan Digital Sky Survey, we have assembled the largest sample available of AGNs which have adequate spectra in both the optical and ultraviolet bands to measure the narrow-line ratio [N ii]/H alpha and also, in the same objects, the broad-line N v/C iv ratio. These permit the measurement of the metallicities in the NLRs and BLRs in the same objects. We find that neither the BLR nor the NLR metallicity correlate with black hole masses or Eddington ratios, but there is a strong correlation between NLR and BLR metallicities. This metallicity correlation implies that outflows from BLRs carry metal-rich gas to NLRs at characteristic radial distances of similar to 1.0 kpc. This chemical connection provides evidence for a kinetic feedback of the outflows to their hosts. Metals transported into the NLR enhance the cooling of the ISM in this region, leading to local star formation after the AGNs turn to narrow-line low-ionization nuclear emission-line regions. This post-AGN star formation is predicted to be observable as an excess continuum emission from the host galaxies in the near-infrared and ultraviolet, which needs to be further explored.