The introductory part of the problem asked to you determine to which sequence belongs the square of every term in B and the square of every term in C. Also, recall
HallsofIvy's post, which shows to which sequences belong squares of numbers. Here I suggest determining to which sequences belongs the fourth powers of numbers.
Here is a start. Since all sequences A - E together cover all numbers, you need to consider five cases. The nth term of, say, sequence C has the form 3 + 5(n - 1). Let k denote n - 1; then the form is 3 + 5k. After some thought it becomes clear that (3 + 5k)(3 + 5k)(3 + 5k)(3 + 5k) = 3^4 + 5(...) where ... denotes some big expression. Also, 3^4 = 81 = 1 + 5 * 16. So, (3 + 5k)^4 = 1 + 5 * j for some j. It is easy but tedious to see that j is a nonnegative integer, so 1 + 5 * j = 1 + 5(j' - 1) where j' = j + 1 ≥ 1. Thus, (3 + 5k)^4 belongs to sequence B. In all this, it is only important to find the remainder of 3^4 when divided by 5; all other terms resulting from expansion of (3 + 5k)^4 are multiples of 5 and thus don't influence to which sequence (3 + 5k)^4 belongs.
Following this pattern, for each sequence and each element in that sequence determine to which sequence belongs the fourth power of that element. In other words, for each positive integer x, take the remainder when x is divided by 5 (it determines the sequence to which x belongs) and use it to determine the remainder when x^4 is divided by 5 (i.e., to which sequence x^4 belongs). Then do the same with 2y^4 for each y. Finally, do this for x^4 + 2y^4. Remember that all that matters is the remainder when divided by 5.
Edit: "remainder of 3 when divided by 5" -> "remainder of 3^4 when divided by 5".