The Dell Inspiron 380 originally used just a 300 watt power supply with no 6-pin PCI Express power connector. The R7 370 requires a higher wattage power supply than that. So no, unfortunately it won't work in an Inspiron 580.
If you have already replaced the power supply with a 620-watt unit, then the R7 370 should work fine, provided you have adequate PCI Express power connectors for the card.
I did, you said you had a 620watt PSU. I said you would be fine but use a PSU calculator to be sure?
No one can come around and do it for you I'm afraid. Good to see my time was well used.