How many rational roots does x^4+3x^3+3x^2+3x+2=0 have?

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?