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?

31 is prime. and it’s prime factorization is just 31.