Powering arduino and solenoids with 12 volt power supply

I have a 12 volt 5 amp power supply that I salvaged from an old projector. I have eight 12 volt 350 mA solenoids that I need to power as well as the Arduino Uno. Is there a way I can use this one power supply to power both the arduino and the solenoids? Can I do so and still utilize the barrel power jack on the arduino (because it happens to fit and would be really nice not to have find another part)

I'm using a 12v 2A power supply to drive a decent sized solenoid and it's working ok. I haven't figured out how to power the arduino from the walwart power AND the solenoid tho...