Factoring by grouping is a method used to factor polynomials, especially useful when you have four terms. Here is how you can factor by grouping step- by-step:

Steps to Factor by Grouping

1. Look for a Greatest Common Factor (GCF) among all terms
   First, check if there is a GCF common to all terms. If yes, factor it out.

2. Group the terms into pairs
   Divide the polynomial into two groups, usually the first two terms and the last two terms.

3. Factor out the GCF from each group
   Find the GCF of each group and factor it out.

4. Look for a common binomial factor
   After factoring each group, you should have a common binomial factor. Factor this binomial out.

5. Write the final factored form
   The result will be the product of the common binomial and the remaining binomial from the groups.

Example

Factor the polynomial 2x2+8x+3x+122x^2+8x+3x+122x2+8x+3x+12:

• Group terms: (2x2+8x)+(3x+12)(2x^2+8x)+(3x+12)(2x2+8x)+(3x+12)
• Factor out GCF in each group: 2x(x+4)+3(x+4)2x(x+4)+3(x+4)2x(x+4)+3(x+4)
• Factor out common binomial (x+4)(x+4)(x+4): (x+4)(2x+3)(x+4)(2x+3)(x+4)(2x+3)

So, 2x2+8x+3x+12=(x+4)(2x+3)2x^2+8x+3x+12=(x+4)(2x+3)2x2+8x+3x+12=(x+4)(2x+3)

Additional Notes

• If the polynomial has three terms (a trinomial), sometimes you can split the middle term into two terms so that the polynomial becomes four terms, then apply grouping

• This method often applies when the coefficient of the squared term is greater than 1 or when the polynomial has four terms.
• After factoring by grouping, check your work by expanding the factors to ensure they multiply back to the original polynomial

This approach simplifies factoring complex polynomials by breaking them into manageable parts.