I have a 08 Sportsman 500HO which the motor starts cutting out between 3600-4600 RPM's

When was the last time it ran.

Some sort of a large parasitic draw from transmission.

Old fuel, dirty carb jets. Low fuel level in carb. Has it been sitting up? Check air filter and plug.

if thats the 2 stroke (pretty sure it is) you have to watch the crank seals on those. cause running probs and eventually destroy the motor.

Replace spark plug (gap correctly),Drain all fuel, replace fuel filter, replenish tank with fresh fuel, do a thorough readjustment of the carb settings (last resort - remove and rebuild carb)

check transmission shift levers.
polaris has over ride safety switches..
owners manual may help....
or fuel ice blockage at or near the tank pickup,or flow/noflow valve.
could be ice in float bowl of carb. (if not efi.)
...maybe use hair dryer to heat area near carb/efi unit.

You need three things to make any gas engine run, in this order double check make sure you have proper timing set, spark and fuel. Could be bad gas or a bad ground from motor to frame.

did you check the float height?

Listen I have a 2009 EFI Polaris sportsman 500 it starts up goes fine down the road but sometimes it takes a fit an starts to bog out when it starts to do that all I do is shut the key off an back on real quick and it takes off again I'm really confused on what is making it do that because I changed the gas an air filter plus put a new spark plug in it also a new battery can anyone tell me what I could do to try an fix this problem it's got to be something easy because when it does that all I have to do is shut the key off an back on than it straightens up weird I know lol but please help me with some suggestions thanks

All most positive it is in your carb take the bowl off it as you take it off there is a small tube that hangs down that all most reaches the bottom of the bowl need to take a small wire and push through it it is blocked .symptoms backfiring no idle coughing and sputtering if it does start