The rational root theorem tells you possible rational roots will be of the form ...
... ±(divisor of 3)/(divisor of 2)
so will be from the set {-3, -3/2, -1, -1/2, 1/2, 1, 3/2, 3}.
By adding coefficients in various ways, we can find
... f(0) = -3
... f(1) = -12
... f(-1) = 8
so there is a root between -1 and 0. When we try -1/2, we find it is a root and we get a quotient of the cubic
... x^3 -3x^2 +x -3.
The pattern in the coefficients tells us we can factor by grouping to get
... x^2(x -3) + (x -3)
So the factorization of f(x) is
... f(x) = (2x +1)(x -3)(x^2 +1)
The last factor has zeros of ±√(-1) = ±i.
The zeros of f(x) are {-1/2, 3, -i, +i}.