<PRE><B>Question 593670</B>
I&#39;m guessing you want to factor this. Let me know if my assumption is correct or not.




{{{6x^4-18x^3+12x^2}}} Start with the given expression.



{{{6x^2(x^2-3x+2)}}} Factor out the GCF {{{6x^2}}}.



Now let&#39;s try to factor the inner expression {{{x^2-3x+2}}}



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Looking at the expression {{{x^2-3x+2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-3}}}, and the last term is {{{2}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{2}}} to get {{{(1)(2)=2}}}.



Now the question is: what two whole numbers multiply to {{{2}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-3}}}?



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 {{{2}}} (the previous product).



Factors of {{{2}}}:

1,2

-1,-2



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{2}}}.

1*2 = 2
(-1)*(-2) = 2


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-3}}}:



&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;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+2=3&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1&lt;/font&gt;&lt;/td&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;-1+(-2)=-3&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-1}}} and {{{-2}}} add to {{{-3}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{-2}}} both multiply to {{{2}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-3}}}



Now replace the middle term {{{-3x}}} with {{{-x-2x}}}. Remember, {{{-1}}} and {{{-2}}} add to {{{-3}}}. So this shows us that {{{-x-2x=-3x}}}.



{{{x^2+highlight(-x-2x)+2}}} Replace the second term {{{-3x}}} with {{{-x-2x}}}.



{{{(x^2-x)+(-2x+2)}}} Group the terms into two pairs.



{{{x(x-1)+(-2x+2)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-1)-2(x-1)}}} Factor out {{{2}}} 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-2)(x-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}



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So {{{6x^2(x^2-3x+2)}}} then factors further to {{{6x^2(x-2)(x-1)}}}



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Answer:



So {{{6x^4-18x^3+12x^2}}} completely factors to {{{6x^2(x-2)(x-1)}}}.



In other words, {{{6x^4-18x^3+12x^2=6x^2(x-2)(x-1)}}}.



Note: you can check the answer by expanding {{{6x^2(x-2)(x-1)}}} to get {{{6x^4-18x^3+12x^2}}} or by graphing the original expression and the answer (the two graphs should be identical).
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