<PRE><B>Question 299372</B>


{{{3x^2+12x-36}}} Start with the given expression.



{{{3(x^2+4x-12)}}} Factor out the GCF {{{3}}}.



Now let&#39;s try to factor the inner expression {{{x^2+4x-12}}}



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Looking at the expression {{{x^2+4x-12}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{4}}}, 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 {{{4}}}?



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 {{{4}}}:



&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)=-11&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)=-4&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)=-1&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=11&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=4&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=1&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 {{{4}}} (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 {{{4}}}



Now replace the middle term {{{4x}}} with {{{-2x+6x}}}. Remember, {{{-2}}} and {{{6}}} add to {{{4}}}. So this shows us that {{{-2x+6x=4x}}}.



{{{x^2+highlight(-2x+6x)-12}}} Replace the second term {{{4x}}} 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 {{{3(x^2+4x-12)}}} then factors further to {{{3(x+6)(x-2)}}}



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Answer:



So {{{3x^2+12x-36}}} completely factors to {{{3(x+6)(x-2)}}}.



In other words, {{{3x^2+12x-36=3(x+6)(x-2)}}}.



Note: you can check the answer by expanding {{{3(x+6)(x-2)}}} to get {{{3x^2+12x-36}}} or by graphing the original expression and the answer (the two graphs should be identical).
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