<PRE><B>Question 490667</B>


{{{36x^2-24xy+4y^2}}} Start with the given expression.



{{{4(9x^2-6xy+y^2)}}} Factor out the GCF {{{4}}}.



Now let&#39;s try to factor the inner expression {{{9x^2-6xy+y^2}}}



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Looking at the expression {{{9x^2-6xy+y^2}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{-6}}}, and the last coefficient is {{{1}}}.



Now multiply the first coefficient {{{9}}} by the last coefficient {{{1}}} to get {{{(9)(1)=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 {{{-6}}}?



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 = 9
3*3 = 9
(-1)*(-9) = 9
(-3)*(-3) = 9


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-6}}}:



&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+9=10&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=6&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)=-10&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3+(-3)=-6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-3}}} and {{{-3}}} add to {{{-6}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{-3}}} both multiply to {{{9}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-6}}}



Now replace the middle term {{{-6xy}}} with {{{-3xy-3xy}}}. Remember, {{{-3}}} and {{{-3}}} add to {{{-6}}}. So this shows us that {{{-3xy-3xy=-6xy}}}.



{{{9x^2+highlight(-3xy-3xy)+y^2}}} Replace the second term {{{-6xy}}} with {{{-3xy-3xy}}}.



{{{(9x^2-3xy)+(-3xy+y^2)}}} Group the terms into two pairs.



{{{3x(3x-y)+(-3xy+y^2)}}} Factor out the GCF {{{3x}}} from the first group.



{{{3x(3x-y)-y(3x-y)}}} Factor out {{{-y}}} from the second group.



{{{(3x-y)(3x-y)}}} Factor out {{{3x-y}}}



{{{(3x-y)^2}}} Condense




So {{{9x^2-6xy+y^2}}} factors to {{{(3x-y)^2}}}



This then means that {{{4(9x^2-6xy+y^2)}}}  becomes {{{4(3x-y)^2}}} 



So the final answer is {{{4(3x-y)^2}}}
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