Graph the equation to get some valuable insights about the factors. The results you get from a graph are not always reliable, but they can speed up the factoring process. The basic idea is that the places where the graphed curve crosses the "X" axis represent a root of the polynomial. If the curve crosses the "X" axis at point "r," "X -- r" is a factor of the polynomial. Sometimes graphs are hard to read and behave strangely when there are multiple roots, but if you have access to a graphing calculator or graphing software, it can definitely speed up the factoring process.
The candidate binomial factors of a polynomial can be generated from the first and last numbers in the polynomial. For example, the fourth degree polynomial 2X^4 + 5X^3 - 5X^2 - 5X + 3 has first number 2 with factors 1 and 2 and last number 3 with factors 1 and 3. The candidate factors are X - 1, X + 1, X - 3, X + 3, 2X - 1, 2X + 1, 2X - 3 and 2X + 3. Try dividing the candidates into the polynomial and see which have no remainders to find the factors of 2X^4 + 5X^3 -5X^2 - 5X + 3.
If 2X^4 + 5X^3 - 5X^2 - 5X + 3 = (X - 1)(X + 1)(X + 3)(2x - 1), the solutions to the four factors are also roots of the fourth degree polynomial. The solution of X - 1 = 0 is X = 1. The solution of X + 1 = 0 is X = -1. The solution of X + 3 = 0 is X = -3. The solution of 2X - 1 = 0 is X = 1/2. This means that the roots of 2X^4 + 5X^3 -5X^2 - 5X + 3 are 1, -1, -3 and 1/2.
The degree of a fourth degree polynomial implies that there are four roots and four factors. If you find less than four roots, some of the roots may be multiples. For example, X^4 - 6X^3 + 13X^2 - 12X + 4 = (X^2 - 4X + 4)(X - 1)^2, so X - 1 is a multiple factor. Another possibility is that there are complex roots, which always come in pairs. Complex roots are indicated when the graphed curve does not cross the "X" axis. A fourth degree polynomial may have two pairs of complex roots and the graphed curve of this polynomial will not cross the "X" axis at all.