The max number of roots that x^4+3x^3+3x^2+3x+2=0 could have is four, based upon the highest power of x (x^4).
Possible rational roots would be formed from the coefficient 1 of x^4 and the coefficient 2 of x^0:
1, -1, 2, -2 (which are the same as 1/1, -1/1, 2/1, -2/1).
Applying synthetic division shows that -1 is a root, leaving a quotient of
x^3 + 2x^2 + x + 2. Similarly, it can be shown that -2 is a root; the quotient is x^2 + 1. The roots of x^2 + 1 = 0 are plus and minus i.
Thus, the roots are {-1, -2, i, -i}, which could also be written as
{-1/1, -2/1, i/1, -i/1).
-1/1 and -2/1 are definitely rational roots. Can -i/1 and i/1 be both imaginary and rational?