Potato and potatoe are both correct. Potato is the currently preferred version, but the flash card at the Dan Quayle event used the other spelling as the standard.
Potatoe is not correct because we no longer live in the 1300s when standard spelling didn’t exist. Potato is the correct spelling, and the way you would want teachers to teach your kids to spell it.
And to think that today we have a president who while may be able to spell does not know how many states there are in the Union. :-))