What is the best way to wash/dry a synthetic down sleeping bag?
Your sleeping bag should have a care label, if not, check the instruction manual. The manufacturer may have more information on their website.
Generally, one of the large washers at the laundry-mat will accommodate most bags. Be sure and use the gentle cycle with a cold water and a very gentle detergent, such as Woolite.
Be careful when drying, it should only be dried on low heat. Some synthetic fabrics can actually melt under high heat.
Dry cleaning may be your only option depending on the fabric used for the shell.