Figured this needs to broken down case by case, where case one is whenever the permutation is even, and case two is when it is odd. Whenever a permutation is odd, then the product of that permutation with a 4-cycle is even, and so I figure we need only show this for whenever it is even.
My other way would be to show that each transposition can be written as the product of four cycles, but I'm a bit lost on this, since four-cycle is itself the product of three transpositions. This was the way to do it.
We just need to show that $(12)$ is a product of 4-cycles; any other transposition can be obtained by swapping numbers, and $S_n$ is generated by transpositions. $$(12)=(1325)(1342)(2543)$$