<PRE><B>Question 614323</B>


{{{-36m^2+6m+12}}} Start with the given expression.



{{{-6(6m^2-m-2)}}} Factor out the GCF {{{-6}}}.



Now let&#39;s try to factor the inner expression {{{6m^2-m-2}}}



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Looking at the expression {{{6m^2-m-2}}}, we can see that the first coefficient is {{{6}}}, the second coefficient is {{{-1}}}, and the last term is {{{-2}}}.



Now multiply the first coefficient {{{6}}} by the last term {{{-2}}} to get {{{(6)(-2)=-12}}}.



Now the question is: what two whole numbers multiply to {{{-12}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-1}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{-12}}} (the previous product).



Factors of {{{-12}}}:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-12}}}.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-1}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-12)=-11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-6)=-4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3+(-4)=-1&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+12=11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+6=4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+4=1&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{-4}}} add to {{{-1}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{-4}}} both multiply to {{{-12}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-1}}}



Now replace the middle term {{{-1m}}} with {{{3m-4m}}}. Remember, {{{3}}} and {{{-4}}} add to {{{-1}}}. So this shows us that {{{3m-4m=-1m}}}.



{{{6m^2+highlight(3m-4m)-2}}} Replace the second term {{{-1m}}} with {{{3m-4m}}}.



{{{(6m^2+3m)+(-4m-2)}}} Group the terms into two pairs.



{{{3m(2m+1)+(-4m-2)}}} Factor out the GCF {{{3m}}} from the first group.



{{{3m(2m+1)-2(2m+1)}}} Factor out {{{2}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(3m-2)(2m+1)}}} Combine like terms. Or factor out the common term {{{2m+1}}}



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So {{{-6(6m^2-m-2)}}} then factors further to {{{-6(3m-2)(2m+1)}}}



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Answer:



So {{{-36m^2+6m+12}}} completely factors to {{{-6(3m-2)(2m+1)}}}.



In other words, {{{-36m^2+6m+12=-6(3m-2)(2m+1)}}}.



Note: you can check the answer by expanding {{{-6(3m-2)(2m+1)}}} to get {{{-36m^2+6m+12}}} or by graphing the original expression and the answer (the two graphs should be identical).


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