<PRE><B>Question 610228</B>


{{{-12x^2-28x+24}}} Start with the given expression.



{{{-4(3x^2+7x-6)}}} Factor out the GCF {{{-4}}}.



Now let&#39;s try to factor the inner expression {{{3x^2+7x-6}}}



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Looking at the expression {{{3x^2+7x-6}}}, we can see that the first coefficient is {{{3}}}, the second coefficient is {{{7}}}, and the last term is {{{-6}}}.



Now multiply the first coefficient {{{3}}} by the last term {{{-6}}} to get {{{(3)(-6)=-18}}}.



Now the question is: what two whole numbers multiply to {{{-18}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{7}}}?



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 {{{-18}}} (the previous product).



Factors of {{{-18}}}:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-18}}}.

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{7}}}:



&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;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-18)=-17&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-9)=-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;3&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;3+(-6)=-3&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;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+18=17&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+9=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;-3&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;-3+6=3&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{9}}} add to {{{7}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{9}}} both multiply to {{{-18}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{7}}}



Now replace the middle term {{{7x}}} with {{{-2x+9x}}}. Remember, {{{-2}}} and {{{9}}} add to {{{7}}}. So this shows us that {{{-2x+9x=7x}}}.



{{{3x^2+highlight(-2x+9x)-6}}} Replace the second term {{{7x}}} with {{{-2x+9x}}}.



{{{(3x^2-2x)+(9x-6)}}} Group the terms into two pairs.



{{{x(3x-2)+(9x-6)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(3x-2)+3(3x-2)}}} Factor out {{{3}}} 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+3)(3x-2)}}} Combine like terms. Or factor out the common term {{{3x-2}}}



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So {{{-4(3x^2+7x-6)}}} then factors further to {{{-4(x+3)(3x-2)}}}



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Answer:



So {{{-12x^2-28x+24}}} completely factors to {{{-4(x+3)(3x-2)}}}.



In other words, {{{-12x^2-28x+24=-4(x+3)(3x-2)}}}.



Note: you can check the answer by expanding {{{-4(x+3)(3x-2)}}} to get {{{-12x^2-28x+24}}} or by graphing the original expression and the answer (the two graphs should be identical).


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