Use your inductive assumption.
You want to see if the statement is true for .
Well you seem to have correctly observed that the expression for is just the expression for with added on.
Since by inductive hypothesis, 16 divides the expression for , all you have to do is verify that 16 divides .
Expand this and you will see every term is divisible by 16. (you may find it easy to expand if you take advantage of its form as a difference of squares).
In your proof, you might want to explain that if a number (16) divides two numbers, then it divides their sum as well. Presumably, this property has been discussed/proved in your class or textbook by now, so just mentioning the fact will probably suffice.