Factor a binomial where the leading coefficient is one, like Z^2 + 8Z +15 or X^2 - 1, by generating some candidates using the constant term. For example, to factor Z^2 + 8Z +15, consider the factors of the constant term 15. These factors are 1, 3, 5, and 15. These factors generate the divisor candidates Z - 1, Z + 1, Z - 3, Z + 3, Z - 5, Z + 5, Z - 15 and Z + 15. If you try each of these, you will find that both Z + 3 and Z + 5 divide the binomial so the factoring of the polynomial is Z^2 + 8Z +15 = (Z + 3)(Z - 5).
Find the candidate divisors for binomials with a leading coefficient greater than one, like 2X^2 +17X - 9, in a slightly more complicated fashion involving the factors of both the leading coefficient and the constant term. The factors of the leading coefficient of 2X^2 + 17X - 9 are 1 and 2, and the factors of the constant term are 1, 3 and 9. The candidates for divisors are: X - 1, X + 1, X - 3, X + 3, X - 9, X + 9, 2X - 1, 2X + 1, 2X - 3, 2X + 3, 2X - 9 and 2X + 9. Trying all of these we find that 2X^2 +17X - 9 = (2X -1)(X + 9).
Use the quadratic formula to find factors when you cannot find monomial divisor by simpler factoring techniques. The quadratic formula actually finds roots, but the roots can tell you the factors. If the roots of aZ^2 + bZ + c are r1 and r2, the binomial can be factored this way: aZ^2 + bZ + c = (Z - r1)(Z - r2). The quadratic formula says that if the binomial is aZ^2 + bZ + c, the roots are (-b + (b^2 - 4ac)^0.5)/2a and (-b - (b^2 - 4ac)^0.5)/2a.