Since the X320 is used, you also want a look under the deck to see the condition. Caked grass or the chute condition can tell how the tractor was kept. Are the blades worn from sharpening and need replacement? Were the spindles greased?
Test the mowing deck thoroughly. Raise and lower the deck from 1 inch, to 2.5 inches to 4 inches. At each height, start the blades and listen for noise. The different heights will put different strains on the belt/PTO system. A bad pulley or out of round sheave may cause belt vibration that should be audible. May want to repeat at mid and fast idle. If you can, test mow to put the blades under load. Sometimes problems only show up under full load.