<PRE><B>Question 168692</B>


Looking at the expression {{{4m^2-8m+3}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{-8}}}, and the last term is {{{3}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{3}}} to get {{{(4)(3)=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 {{{-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 {{{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
2*6
3*4
(-1)*(-12)
(-2)*(-6)
(-3)*(-4)


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=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=13&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=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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+4=7&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)=-13&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+(-6)=-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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-4)=-7&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-6}}} add to {{{-8}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-6}}} both multiply to {{{12}}} &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 {{{-2m-6m}}}. Remember, {{{-2}}} and {{{-6}}} add to {{{-8}}}. So this shows us that {{{-2m-6m=-8m}}}.



{{{4m^2+highlight(-2m-6m)+3}}} Replace the second term {{{-8m}}} with {{{-2m-6m}}}.



{{{(4m^2-2m)+(-6m+3)}}} Group the terms into two pairs.



{{{2m(2m-1)+(-6m+3)}}} Factor out the GCF {{{2m}}} from the first group.



{{{2m(2m-1)-3(2m-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.



{{{(2m-3)(2m-1)}}} Combine like terms. Or factor out the common term {{{2m-1}}}


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Answer:



So {{{4m^2-8m+3}}} factors to {{{(2m-3)(2m-1)}}}.



Note: you can check the answer by FOILing {{{(2m-3)(2m-1)}}} to get {{{4m^2-8m+3}}} or by graphing the original expression and the answer (the two graphs should be identical).
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