Graph the polynomial. The places where the graphed curve crosses the X axis represent roots --- values of the variable that makes the polynomial equal zero. If p is such a point then X - p is a factor of the polynomial. The graph can provide insight into how many roots to expect and how many of these roots are multiples. For example, the graph of X^4 - 4X^3 + 10X^2 - 12X + 5 does not cross the X axis, but it does touch the X axis at the point (1,0). This suggests that 1 is a root and (X -1)^2 is a factor. It also suggests that there are two complex factors. Complex factors are not used in practical application problems.
Find the candidate factors. Look at the first and last numbers in the polynomial. The only possible factors of the polynomial will consist of expressions whose first and last numbers are factors of the first and last factors of the polynomial. For example, the candidate factors of the polynomial 2X^2 - 14X + 20 are expressions whose first number is a factor of 2 and whose last number is a factor of 20. These candidates are X - 1, X + 1, X - 2, X + 2, X - 5, X + 5, X - 10, X + 10, X - 20, X + 20, 2X - 1, 2X + 1, 2X - 2, 2X + 2, 2X - 5, 2X + 5, 2X - 10, 2X + 10, 2X - 20 and 2X + 20.
Try the candidates until the polynomial is factored, such as 2X^2 - 14X + 20 = (2X - 4)(X - 5). Setting each factor to zero and solving, you'll find that X = 2 and X = 5 are both roots for this polynomial.