<PRE><B>Question 725324</B>


{{{x^3+8x^2+12x}}} Start with the given expression.



{{{x(x^2+8x+12)}}} Factor out the GCF {{{x}}}.



Now let&#39;s try to factor the inner expression {{{x^2+8x+12}}}



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Looking at the expression {{{x^2+8x+12}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{8}}}, and the last term is {{{12}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{12}}} to get {{{(1)(12)=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 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12


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=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;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;/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 {{{8x}}} with {{{2x+6x}}}. Remember, {{{2}}} and {{{6}}} add to {{{8}}}. So this shows us that {{{2x+6x=8x}}}.



{{{x^2+highlight(2x+6x)+12}}} Replace the second term {{{8x}}} with {{{2x+6x}}}.



{{{(x^2+2x)+(6x+12)}}} Group the terms into two pairs.



{{{x(x+2)+(6x+12)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+2)+6(x+2)}}} Factor out {{{6}}} 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+6)(x+2)}}} Combine like terms. Or factor out the common term {{{x+2}}}



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So {{{x(x^2+8x+12)}}} then factors further to {{{x(x+6)(x+2)}}}



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Answer:



So {{{x^3+8x^2+12x}}} completely factors to {{{x(x+6)(x+2)}}}.



In other words, {{{x^3+8x^2+12x=x(x+6)(x+2)}}}.



Note: you can check the answer by expanding {{{x(x+6)(x+2)}}} to get {{{x^3+8x^2+12x}}} or by graphing the original expression and the answer (the two graphs should be identical).
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