Graph the fifth-order polynomial on a graphing calculator. The places where the graphed curve crosses the X axis represent real valued roots of the polynomial. If p is such a point, the monomial X - p is a factor of the polynomial. For example, if the graph crosses the X axis at +3, X - 3 is a factor of the polynomial. You should note that if the curve crosses the X axis at 2/3 it is more likely that the monomial factor is 3X - 2 than it is that the factor is X - 2/3. Same root, different monomials.
Find candidates for monomial factors of the polynomial, consider all combinations of the factors of the first and last number in the polynomial. For example, for the fifth-order polynomial 2X^5 - 5X^4 - 6X^3 + 8X^2 + 4X + 3, the first number is 2 which has factors 1 and 2, and the last number is 3 which has factors 1 and 3. Candidate monomial divisors include X - 1, X + 1, X - 3, X + 3, 2X -1, 2X + 1, 2X - 3 and 2X + 3. Trying these one at a time, we find that X -1, X + 1, X - 3 and 2X + 1 all divide the polynomial.
The tricky part of this factoring is that this is only four factors and there should be 5 because it is a fifth order polynomial. Complex roots always come in pairs so the missing factor cannot be complex. A little investigation reveals that (X + 1) is a multiple root so 2X^5 - 5X^4 - 6X^3 + 8X^2 + 4X + 3 = (X - 1)(X + 1)(X + 1)(X - 3)(2X + 1).