There's *probably* nothing stopping you setting up a dual boot system .. but to be 100% sure, you'll need to create a LiveUSB stick, bnoot to it, and send the output from a couple of commands for us.
If you're unsure how to create a LiveUSB of Peppermint 4 .. see "Stage 1" here:http://linuxforums.org.uk/index.php?topic=11109.0
DO NOT install Peppermint yet .. just create the LiveUSB and make sure you can boot to it selecting "Try Peppermint".
The reason I say don't install yet is I want to see the partition layout, and see if we'd be better off keeping the Windows bootloader .. so you don't loose access to the recovery partition.
Let us know when you've created and can boot the LiveUSB, and we'll go from there.....
Any questions, just ask
BTW .. Welcome to the forum