Here are the steps required for factoring a trinomial when the leading coefficient is not 1:

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.
Step 2 : Decide if the three 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 3 : Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.
Step 4 : List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x.
Step 5 : After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3.
Step 6 : Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.
Step 7 : Now that the problem is written with four terms, you can factor by grouping.

Example 1 – Factor: Example 1

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order.
Step 1
Step 2: Decide if the three 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 three terms only have a 1 in common which is of no help.
Step 2
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 6 and –2.
Step 3
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 3 and 4 can combine to equal 1.
Step 4
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, –3 and +4 combine to equal +1 and –3 times +4 is –12.
Step 5
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.
Step 6
Step 7: Now that the problem is written with four terms, you can factor by grouping.
Step 7

Example 2 – Factor: Example 2

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order.
Step 1
Step 2: Decide if the three 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 three terms only have a 1 in common which is of no help.
Step 2
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 12 and 15.
Step 3
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 9 and 20 can combine to equal 29.
Step 4
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, –9 and –20 combine to equal –29 and –9 times –20 is 180.
Step 5
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.
Step 6
Step 7: Now that the problem is written with four terms, you can factor by grouping.
Step 7

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Example 3 – Factor: Example 3

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem needs to be rewritten as:
Step 1
Step 2: Decide if the three 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 three terms have a 2 in common, which leaves:
Step 2
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 3 and –8.
Step 3
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 4 and 6 can combine to equal 2.
Step 4
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, +4 and –6 combine to equal –2 and +4 times –6 is –24.
Step 5
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. Do not forget to include 2 (the GCF) as part of your answer.
Step 6
Step 7: Now that the problem is written with four terms, you can factor by grouping. Do not forget to include 2 (the GCF) as part of your final answer.
Step 7

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Example 4 – Factor: Example 4

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order.
Step 1
Step 2: Decide if the three 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 three terms have a 3x in common, which leaves:
Step 2
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 12 and 2.
Step 3
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 3 and 8 can combine to equal 11.
Step 4
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, +3 and +8 combine to equal +11 and +3 times +8 is 24.
Step 5
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. Do not forget to include 3x (the GCF) as part of your answer.
Step 6
Step 7: Now that the problem is written with four terms, you can factor by grouping. Do not forget to include 2 (the GCF) as part of your final answer.
Step 7

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Note: In each example above you could not use the shortcut because the leading coefficient was not a 1.