To consider congruences modulo 5 is to consider all the remainders,

When we raise to the exponent of 4 the forms take respectively,

You might have realized that is Fermat's Little Theorem.

But anyways the two possible forms of (a fourth power) are: .

Thus, the possibilities are in the Diophantine Equation:

---> Impossible, RHS different form.

---> Impossible

---> Possible

---> Impossible.

Thus, what your homework was asking was not correct.

The second case, is still possible.