<PRE><B>Question 481981</B>


{{{40k^2-112k+24}}} Start with the given expression.



{{{8(5k^2-14k+3)}}} Factor out the GCF {{{8}}}.



Now let&#39;s try to factor the inner expression {{{5k^2-14k+3}}}



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Looking at the expression {{{5k^2-14k+3}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-14}}}, and the last term is {{{3}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{3}}} to get {{{(5)(3)=15}}}.



Now the question is: what two whole numbers multiply to {{{15}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-14}}}?



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 {{{15}}} (the previous product).



Factors of {{{15}}}:

1,3,5,15

-1,-3,-5,-15



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{15}}}.

1*15 = 15
3*5 = 15
(-1)*(-15) = 15
(-3)*(-5) = 15


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-14}}}:



&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;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+15=16&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;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+5=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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-15)=-16&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;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-5)=-8&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that there are no pairs of numbers which add to {{{-14}}}. So {{{5k^2-14k+3}}} cannot be factored.



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&lt;a name=&quot;ans&quot;&gt;


Answer:



So {{{40k^2-112k+24}}} simply factors to {{{8(5k^2-14k+3)}}}



In other words, {{{40k^2-112k+24=8(5k^2-14k+3)}}}.
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