Suppose a and b are roots of a fourth degree polynomial.
Then, (x - a) and (x - b) will be the factors of the polynomial and hence, (x-a)(x-b) will be a factor of the polynomial.
Now, to find the other two factors, divide the polynomial with (x-a)(x-b) = x2 - x(a+b) + ab. You will get 0 as the remainder and the quotient will be a second degree polynomial.
By division algorithm,
p(x) = q(x) (x-a)(x-b)
Hence, the roots of q(x) will be the remaining two roots of the fourth degree polynomial.
Example: If two of the zeroes of the polynomial p(x) = 5x4 - 5x3 - 33x2 + 3x + 18 are, find the other two zeroes.