Factoring completely is an algebraic process that incorporates the three common factoring techniques: the greatest common factor, trinomials and the difference between two squares. These techniques by themselves often factor a polynomial into a single pattern, while factoring completely utilizes them together to simplify an expression as much as possible.

**Factor the greatest common factor from the expression**In the example, 12x^4 - 3x^2 - 54, the greatest common factor is 3. Therefore, the 3 is removed from the initial equation, and the expression becomes 3(4x^4 - x^2 - 18). If there is no greatest common factor to be found, this step may be skipped.

**Factor a trinomial**If there are more than two terms in an expressions, it may be factored into a trinomial. In the example, 3(4x^4 - x^2 - 18), the expression would become 3(4x^2 - 9)(x^2 + 2) when factored.

**Factor a difference between two squares**Finally, in the example 3(4x^2 - 9)(x^2 + 2), (4x^2 - 9) may be factored as a difference between two squares. The expression then becomes 3(2x + 3)(2x- 3)(x^2 + 2) and is factored completely. Many equations may not be factored as a difference between two squares, in which case, this step may be skipped.