Graphing a polynomial is a good first step in finding its factors. The points where the graphed curve crosses the X axis are roots of the polynomial. If the curve crosses the axis at point p, then p is a root of the polynomial and X - p is a factor of the polynomial. You should check the factors you get from a graph because it is easy to mistake a reading from a graph. It is also easy to miss multiple roots on a graph.
The candidate binomial factors for a polynomial are composed of the combinations of the factors of the first and last numbers in the polynomial. For example 3X^2 - 18X - 15 has as its first number 3, with factors 1 and 3, and as its last number 15, with factors 1, 3, 5 and 15. The candidate factors are X - 1, X + 1, X - 3, X + 3, X - 5, X + 5, X - 15, X + 15, 3X - 1, 3X + 1, 3X - 3, 3X + 3, 3X - 5, 3X + 5, 3X - 15 and 3X + 15.
Trying each of the candidate factors, we find that 3X + 3 and X - 5 divide 3X^2 - 18X - 15 with no remainder. So 3X^2 - 18X - 15 = (3X + 3)(X - 5). Notice that 3X + 3 is a factor that we would have missed if we relied on the graph alone. The curve would cross the X axis at -1, suggesting that X - 1 is a factor. Of course, it really is because 3X^2 - 18X - 15 = 3(X + 1)(X - 5).
Once you have the binomial factors, it is easy to find the roots of a polynomial -- the roots of the polynomial are the same as the roots of the binomials. For example, the roots of 3X^2 - 18X - 15 = 0 are not obvious, but if you know that 3X^2 - 18X - 15 = (3X + 3)(X - 5), the root of 3X + 3 = 0 is X = -1 and the root of X - 5 = 0 is X = 5.