<PRE><B>Question 189872</B>


Looking at the expression {{{3m^2-8m-3}}}, we can see that the first coefficient is {{{3}}}, the second coefficient is {{{-8}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{3}}} by the last term {{{-3}}} to get {{{(3)(-3)=-9}}}.



Now the question is: what two whole numbers multiply to {{{-9}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-8}}}?



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 {{{-9}}} (the previous product).



Factors of {{{-9}}}:

1,3,9

-1,-3,-9



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-9}}}.

1*(-9)
3*(-3)
(-1)*(9)
(-3)*(3)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-8}}}:



&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=red&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;1+(-9)=-8&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-3)=0&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+9=8&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+3=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{1}}} and {{{-9}}} add to {{{-8}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-9}}} both multiply to {{{-9}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-8}}}



Now replace the middle term {{{-8m}}} with {{{m-9m}}}. Remember, {{{1}}} and {{{-9}}} add to {{{-8}}}. So this shows us that {{{m-9m=-8m}}}.



{{{3m^2+highlight(m-9m)-3}}} Replace the second term {{{-8m}}} with {{{m-9m}}}.



{{{(3m^2+m)+(-9m-3)}}} Group the terms into two pairs.



{{{m(3m+1)+(-9m-3)}}} Factor out the GCF {{{m}}} from the first group.



{{{m(3m+1)-3(3m+1)}}} Factor out {{{3}}} 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.



{{{(m-3)(3m+1)}}} Combine like terms. Or factor out the common term {{{3m+1}}}


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Answer:



So {{{3m^2-8m-3}}} factors to {{{(m-3)(3m+1)}}}.



Note: you can check the answer by FOILing {{{(m-3)(3m+1)}}} to get {{{3m^2-8m-3}}} or by graphing the original expression and the answer (the two graphs should be identical).
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