The hints outline exactly what you need to do. Could you explain what trouble you are having?
Consider the ring Z of integers.
(a) Prove that the ideal I = (4) is not a maximal ideal.
Hint. Find an ideal J of Z such that (4) is a subset of J subset of Z but J does not equal (4) and J does not equal Z.
(b) Prove that the ideal I = (3) is a maximal ideal.
Hint. Suppose that J is any ideal of Z such that (3) is a subset of J subset of Z. Prove that J = I or J = Z.