<PRE><B>Question 299482</B>


{{{2x^2-12x+16}}} Start with the given expression.



{{{2(x^2-6x+8)}}} Factor out the GCF {{{2}}}.



Now let&#39;s try to factor the inner expression {{{x^2-6x+8}}}



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Looking at the expression {{{x^2-6x+8}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-6}}}, and the last term is {{{8}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{8}}} to get {{{(1)(8)=8}}}.



Now the question is: what two whole numbers multiply to {{{8}}} (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 {{{8}}} (the previous product).



Factors of {{{8}}}:

1,2,4,8

-1,-2,-4,-8



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{8}}}.

1*8 = 8
2*4 = 8
(-1)*(-8) = 8
(-2)*(-4) = 8


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;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+8=9&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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+4=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;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-8)=-9&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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+(-4)=-6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-4}}} add to {{{-6}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-4}}} both multiply to {{{8}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-6}}}



Now replace the middle term {{{-6x}}} with {{{-2x-4x}}}. Remember, {{{-2}}} and {{{-4}}} add to {{{-6}}}. So this shows us that {{{-2x-4x=-6x}}}.



{{{x^2+highlight(-2x-4x)+8}}} Replace the second term {{{-6x}}} with {{{-2x-4x}}}.



{{{(x^2-2x)+(-4x+8)}}} Group the terms into two pairs.



{{{x(x-2)+(-4x+8)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-2)-4(x-2)}}} Factor out {{{4}}} 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-4)(x-2)}}} Combine like terms. Or factor out the common term {{{x-2}}}



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So {{{2(x^2-6x+8)}}} then factors further to {{{2(x-4)(x-2)}}}



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Answer:



So {{{2x^2-12x+16}}} completely factors to {{{2(x-4)(x-2)}}}.



In other words, {{{2x^2-12x+16=2(x-4)(x-2)}}}.



Note: you can check the answer by expanding {{{2(x-4)(x-2)}}} to get {{{2x^2-12x+16}}} or by graphing the original expression and the answer (the two graphs should be identical).
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