Factoring rewrites a polynomial as a product of simpler expressions. Expansion reverses the operation. Before looking for a special pattern, check whether every term has a common numerical or variable factor. Removing it first often reveals a simpler expression inside the parentheses.
Take out a common factor
The coefficients 12 and 18 share a factor of 6. Both terms also contain at least one x. After dividing each term by 6x, the remaining terms are 2x and 3. Distributing 6x recovers both original terms.
Group terms to expose a repeated expression
With four terms, grouping in pairs can produce the same expression in two places. In this example the first pair has factor x and the second pair has factor 2. Both leave the same sum, which becomes one factor of the final product.
Check the middle term of a perfect square
A trinomial with square first and last terms is a perfect square only if the middle term matches twice the product of their roots with the appropriate sign. Here the middle term is negative, so the square contains a difference.
A difference of squares produces conjugate factors
Subtracting one square from another gives a product of their sum and difference. When expanded, the cross terms cancel. Apply the pattern again if a remaining factor is itself a difference of squares.
Recognize all four terms of a perfect cube
A binomial cube has two interior terms with coefficients based on 3. Check both and track the signs before treating a four-term expression as a cube. Expanding the final result is a direct check.
