For a linear polynomial $p$, you'll always have $p(n+1)−p(n)$ the same. If you write down a table
1 2 3 4 5
p(1) p(2) p(3) p(4) p(5)
which in our case would be this:
1 2 3 4 5
1 -1 1 -1 1
and then write the differences $p(2)−p(1)$, $p(3)−p(2)$, etc in a row beneath, you'd get (again in our case)
1 2 3 4 5
1 -1 1 -1 1
-2 2 -2 2
That new row is called the "first differences". For a linear function, the entries would all be the same. You can also write down second, third and fourth differences:
1 2 3 4 5
1 -1 1 -1 1
-2 2 -2 2
4 -4 4
-8 8
16
For a function with degree 2, the second differences will all be the same.
In our case fourt difference is same. So degree is 4
Answer is $D$