Make a substitution so the exponents of the polynomial are integers, because the factoring algorithms assume that polynomials are non-negative integers. For example, if the equation is X^1/2 = 3X^1/4 - 2, make the substitution Y = X^1/4 to get Y^2 = 3Y - 2 and put this in standard format Y^2 - 3Y + 2 = 0 as a prelude to factoring. If the factoring algorithm produces Y^2 - 3Y + 2 = (Y -1)(Y - 2) = 0, then the solutions are Y = 1 and Y = 2. Because of the substitution, the real roots are X = 1^4 = 1 and X = 2^ 4 = 16.
Put the polynomial with integers in standard form -- the terms have the exponents in descending order. The candidate factors are made from combinations of factors of the first and last numbers in the polynomial. For example, the first number in 2X^2 - 8X + 6 is 2, which has factors 1 and 2. The last number in 2X^2 - 8X + 6 is 6, which has factors 1, 2, 3 and 6. Candidate factors are X - 1, X + 1, X - 2, X + 2, X - 3, X + 3, X - 6, X + 6, 2X - 1, 2X + 1, 2X - 2, 2X + 2, 2X - 3, 2X + 3, 2X - 6 and 2X + 6.
Find the factors, find the roots and undo the substitution. Try the candidates to see which ones divide the polynomial. For example, 2X^2 - 8X + 6 = (2X -2)(x - 3) so the roots are X = 1 and X = 3. If there was a substitution to make the exponents integers, this is the time to undo the substitution.