Here are the steps required for factoring by grouping:

Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.
Step 3: Factor out the GCF from each of the two groups. In the second group, you have a choice of factoring out a positive or negative number. To determine whether you should factor out a positive or negative number, you need to look at the signs before the second and fourth terms. If the two signs are the same (both positive or both negative) you need to factor out a positive number. If the two signs are different, you must factor out a negative number.
Step 4: If the factors inside of the parenthesis are exactly the same, it is time for the 2 for 1 special. The one thing that the two groups have in common should be what is in parenthesis, so you can factor out what is inside the parenthesis, but only write what is inside the parenthesis once. If what is inside the parenthesis does not match, you need to rearrange the four terms and try again until you get a perfect match. If you have rearranged the problems a couple of times and still have not found a perfect match, then the problem does not factor.
Step 5: Determine if the remaining factors can be factored any further.

Example 1 – Factor: Example 1

Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the four terms only have a 1 in common which is of no help.
Step 1
Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.
Step 2
Step 3: Factor out the GCF from each of the two groups. In this problem, the signs in front of the 5x2 and the 15 are different, so you need to factor out a positive 3.
Step 3
Step 4: Notice that what is inside the parenthesis is a perfect match, so it is time for the 2 for 1 special. The one thing that the two groups have in common is (x – 5), so you can factor out (x – 5) leaving the following:
Step 4
Step 5: Determine if any of the remaining factors can be factored further. In this case they can not so the final answer is:
Step 5

Example 2 – Factor: Example 2

Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the four terms only have a 1 in common which is of no help.
Step 1
Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.
Step 2
Step 3: Factor out the GCF from each of the two groups. In this problem, the signs in front of the 20x and the 15y are different, so you need to factor out a negative 3y.
Step 3
Step 4: Notice that what is inside the parenthesis is a perfect match, so it is time for the 2 for 1 special. The one thing that the two groups have in common is (x + 5), so you can factor out (x + 5) leaving the following:
Step 4
Step 5: Determine if any of the remaining factors can be factored further. In this case they can not so the final answer is:
Step 5

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Example 3 – Solve: Example 3

Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the four terms have a 3 in common, which leaves:
Step 1
Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.
Step 2
Step 3: Factor out the GCF from each of the two groups. In this problem, the signs in front of the 2x2 and the 10 are the same, so you need to factor out a positive 5.
Step 3
Step 4: Notice that what is inside the parenthesis is a perfect match, so it is time for the 2 for 1 special. The one thing that the two groups have in common is (x – 2), so you can factor out (x – 2) leaving the following:
Step 4
Step 5: Determine if any of the remaining factors can be factored further. In this case they can not so the final answer is:
Step 5

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Example 4 – Solve: Example 4

Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the four terms only have a 1 in common which is of no help.
Step 1
Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.
Step 2
Step 3: Factor out the GCF from each of the two groups. In this problem, the signs in front of the ab and the bx are different, so you need to factor out a negative x.
Step 3
Step 4: Notice that what is inside the parenthesis is not a perfect match, so you need to rearrange the four terms and try again. When you rearrange the terms, try to follow a pattern when you write them down. In this case notice that the first terms has an x, the second an a, the third an x, and the fourth an a.
Step 4
Step 5: Factor out the GCF from each of the two groups. In this problem, the signs in front of the ax and the ab are different, so you need to factor out a negative b.
Step 5
Step 6: Notice that what is inside the parenthesis is a perfect match, so it is time for the 2 for 1 special. The one thing that the two groups have in common is (x – a), so you can factor out (x – a) leaving the following:
Step 6
Step 7: Determine if any of the remaining factors can be factored further. In this case they can not so the final answer is:
Step 7

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Example 5 – Solve: Example 5

Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the four terms only have a 1 in common which is of no help.
Step 1
Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.
Step 2
Step 3: Factor out the GCF from each of the two groups. In this problem, the signs in front of the 2x2 and the 18 are different, so you need to factor out a negative 9.
Step 3
Step 4: Notice that what is inside the parenthesis is a perfect match, so it is time for the 2 for 1 special. The one thing that the two groups have in common is (x + 2), so you can factor out (x + 2) leaving the following:
Step 4
Step 5: Determine if any of the remaining factors can be factored further. In this case one of the factors is a difference of square, which can be factored into:
Step 5

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