To factor polynomials with four terms, the most common and effective method is called factoring by grouping. Here is a step-by-step guide on how to do it:

How to Factor Polynomials with Four Terms by Grouping

1. Group the terms into two pairs
   Split the polynomial into two groups, each containing two terms. For example, for the polynomial x3+x2−x−1x^3+x^2-x-1x3+x2−x−1, group as (x3+x2)+(−x−1)(x^3+x^2)+(-x-1)(x3+x2)+(−x−1).

2. Factor out the Greatest Common Factor (GCF) from each group
   Find the GCF of each pair and factor it out. Using the example above:

   • From x3+x2x^3+x^2x3+x2, factor out x2x^2x2, giving x2(x+1)x^2(x+1)x2(x+1).
   • From −x−1-x-1−x−1, factor out −1-1−1, giving −1(x+1)-1(x+1)−1(x+1).

3. Look for a common binomial factor
   After factoring, you should have a common binomial factor in both groups. In the example, the common factor is (x+1)(x+1)(x+1).

4. Factor out the common binomial
   Factor the common binomial out of the entire expression:

x2(x+1)−1(x+1)=(x+1)(x2−1)x^2(x+1)-1(x+1)=(x+1)(x^2-1)x2(x+1)−1(x+1)=(x+1)(x2−1)

5. Factor further if possible
   Check if the remaining polynomial can be factored further. In this example, x2−1x^2-1x2−1 is a difference of squares and factors as (x−1)(x+1)(x-1)(x+1)(x−1)(x+1). So the full factorization is:

(x+1)(x−1)(x+1)=(x+1)2(x−1)(x+1)(x-1)(x+1)=(x+1)^2(x-1)(x+1)(x−1)(x+1)=(x+1)2(x−1)

Summary of the Grouping Method

• Split the polynomial into two groups of two terms each.
• Factor out the GCF from each group.
• Identify and factor out the common binomial.
• Factor any remaining expressions if possible.

Example

Factor 2x3−3x2+18x−272x^3-3x^2+18x-272x3−3x2+18x−27:

• Group: (2x3−3x2)+(18x−27)(2x^3-3x^2)+(18x-27)(2x3−3x2)+(18x−27)
• Factor each group: x2(2x−3)+9(2x−3)x^2(2x-3)+9(2x-3)x2(2x−3)+9(2x−3)
• Common binomial: (2x−3)(2x-3)(2x−3)
• Factor out common binomial: (x2+9)(2x−3)(x^2+9)(2x-3)(x2+9)(2x−3)

This method works well for many four-term polynomials and is a foundational technique in algebra