Graph the equation. Each time the graphed curve crosses the x-axis represents a real monomial factor. The place where the curve crosses the x-axis is a root of the equation and x - p, where p is the point where the curve crosses the axis -- is a real valued monomial factor. Complex roots always come in pairs, so the number of real valued monomial factors will be 0, 2 or 4. You cannot really get the factors from the graph, even if there are four of them, but the graph does indicate the type of factors to expect.
Look at the first and last numbers in the quartic equation to find candidates for the polynomial factors. For example, for 2X^4 -13X^3 +28X^2 -23X + 6 the first number is 2, which has factors 1 and 2. The last number is 6 which has factors 1, 2, 3 and 6. The candidates for factors for the quartic 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 + 3,2 X - 6 and 2X + 6. Trying each of these, we find that 2X^4 -13X^3 +28X^2 -23X + 6 = (X - 1)(X - 2)(X - 3)(2X - 1). This quartic has four real roots. If two of the roots were complex, we would have found two monomial divisors. If all of the roots were complex, none of the candidates would be divisors.
Factor the binomial factors using the quadratic formula. For some applications, complex roots are undesirable, so the binomial factors are left un-factored. For example, 4X^4 - X^3 - 2X^2 - 2X + 4 = (X - 1)(X - 2)(X^2 + 2X +2). None of the other monomial candidates divide 4X^4 - X^3 - 2X^2 - 2X + 4. You can use the quadratic formula to factor X^2 + 2X +2 into complex monomials: X^2 + 2X +2 = (X + 1 + i)(X + 1 - i), so 4X^4 - X^3 - 2X^2 - 2X + 4 = (X - 1)(X - 2)(X + 1 + i)(X + 1 - i). The application determines if factoring all the way to complex numbers is required.