The Rational Root Theorem says any rational zeros of f must be a fraction with a factor of the constant (9 here) in the numerator and a factor of the leading coefficient (4 here) in the denominator. We have to include negative factors.
Factors of 9: -9, -3, -1, 1, 3, 9
Factors of 4: 1, 2, 4
We only have to consider negative numbers for the numerator to get all the possible values.
We just make the list, going through all the numerators one denominator at a time.
Possible rational zeros, i.e. possible rational values for x such that f(x)=0 are
Answer: -9, -3, -1, 1, 3, 9, -9/2, -3/2, -1/2, 1/2, 3/2, 9/2, -9/4, -3/4, -1/4, 1/4, 3/4, 9/4