Since the degree of this polynomial is 5, there will be 5 possible zeros. To find the possible rational 0s, use the rational root theorem (p/q). P is the last, non x value, which here it is the four on the end. The q is the leading coefficient, which is also q. Next, find all of the factors of q and p, which since they are both 4, are ±1, ±2, and ±4. Next do all possible values of p/q, which are ±1, ±2, ±4, ±1/2, and ±1/4. These are all your possible rational zeros. complex 0s only come in pairs, so the maximum there can be is 4 complex zeros, meaning there is at least one rational, real 0. (i graphed it it is -1/2, so all others must be rational or imaginary)
