Polynomial equations can be solved by factoring through the following steps. The first step is to write the equation in its correct form, which involves distributing the terms by removing all parentheses on both sides of the equation, combining all the like terms and writing the equation in standard form. The second step is to use factoring methods to factor the problem, after which the Zero Product Principle is applied.

All factors set to zero are solved by setting the X on one side and the answer on the opposite side. The factoring strategies mentioned involve identifying the greatest common factor (GCF) of the polynomial and dividing every term of the polynomial by the GCF. If there is no GCF for all the terms in the polynomial, and the polynomial has four terms, the terms are factored by grouping. The first two terms are grouped together and the same is done to the last two terms.

The next step is to factor out a GCF from each of the binomials, which are then used to factor out a common binomial. The Zero Product Principle is the only guarantee that a solution exists, since the product cannot be zero unless the factors are equated to zero.