<PRE><B>Question 251553</B>
I&#39;m not sure what you mean by the box method, but here&#39;s one way to do it.




Looking at the expression {{{5x^2-42x+16}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-42}}}, and the last term is {{{16}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{16}}} to get {{{(5)(16)=80}}}.



Now the question is: what two whole numbers multiply to {{{80}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-42}}}?



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 {{{80}}} (the previous product).



Factors of {{{80}}}:

1,2,4,5,8,10,16,20,40,80

-1,-2,-4,-5,-8,-10,-16,-20,-40,-80



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{80}}}.

1*80 = 80
2*40 = 80
4*20 = 80
5*16 = 80
8*10 = 80
(-1)*(-80) = 80
(-2)*(-40) = 80
(-4)*(-20) = 80
(-5)*(-16) = 80
(-8)*(-10) = 80


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-42}}}:



&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;80&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+80=81&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;40&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+40=42&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+20=24&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+16=21&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8+10=18&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;-80&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-80)=-81&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;-40&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+(-40)=-42&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-20)=-24&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-16)=-21&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-8+(-10)=-18&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-40}}} add to {{{-42}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-40}}} both multiply to {{{80}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-42}}}



Now replace the middle term {{{-42x}}} with {{{-2x-40x}}}. Remember, {{{-2}}} and {{{-40}}} add to {{{-42}}}. So this shows us that {{{-2x-40x=-42x}}}.



{{{5x^2+highlight(-2x-40x)+16}}} Replace the second term {{{-42x}}} with {{{-2x-40x}}}.



{{{(5x^2-2x)+(-40x+16)}}} Group the terms into two pairs.



{{{x(5x-2)+(-40x+16)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(5x-2)-8(5x-2)}}} Factor out {{{8}}} 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.



{{{(x-8)(5x-2)}}} Combine like terms. Or factor out the common term {{{5x-2}}}



===============================================================



Answer:



So {{{5x^2-42x+16}}} factors to {{{(x-8)(5x-2)}}}.



In other words, {{{5x^2-42x+16=(x-8)(5x-2)}}}.



Note: you can check the answer by expanding {{{(x-8)(5x-2)}}} to get {{{5x^2-42x+16}}} or by graphing the original expression and the answer (the two graphs should be identical).
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