<PRE><B>Question 507228</B>


Looking at the expression {{{n^2+4n-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 {{{4n}}} with {{{-2n+6n}}}. Remember, {{{-2}}} and {{{6}}} add to {{{4}}}. So this shows us that {{{-2n+6n=4n}}}.



{{{n^2+highlight(-2n+6n)-12}}} Replace the second term {{{4n}}} with {{{-2n+6n}}}.



{{{(n^2-2n)+(6n-12)}}} Group the terms into two pairs.



{{{n(n-2)+(6n-12)}}} Factor out the GCF {{{n}}} from the first group.



{{{n(n-2)+6(n-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.



{{{(n+6)(n-2)}}} Combine like terms. Or factor out the common term {{{n-2}}}



===============================================================



Answer:



So {{{n^2+4n-12}}} factors to {{{(n+6)(n-2)}}}.



In other words, {{{n^2+4n-12=(n+6)(n-2)}}}.



Note: you can check the answer by expanding {{{(n+6)(n-2)}}} to get {{{n^2+4n-12}}} or by graphing the original expression and the answer (the two graphs should be identical).


Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at &lt;a href=&quot;mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help&quot;&gt;jim_thompson5910@hotmail.com&lt;/a&gt;
or you can visit my website here: &lt;a href=&quot;http://www.freewebs.com/jimthompson5910/home.html&quot;&gt;http://www.freewebs.com/jimthompson5910/home.html&lt;/a&gt;


Thanks,


Jim </PRE>