Combining several methods of factorization

When factoring a polynomial we follow this order: 1) if there is a common factor, take it out of brackets; 2) try the short multiplication formulas; 3) otherwise try grouping. Sometimes the methods are applied one after another: 2x³ - 18x = 2x(x² - 9) = 2x(x - 3)(x + 3). We continue until the factors cannot be factored further. The new formulas are a³ + b³ = (a + b)(a² - ab + b²), the sum of cubes, and a³ - b³ = (a - b)(a² + ab + b²), the difference of cubes. They can be checked by multiplying out: (a + b)(a² - ab + b²) = a³ + b³. For example 27 + b³ = (3 + b)(9 - 3b + b²). The trinomial in the second bracket cannot be factored further.