<PRE><B>Question 201377</B>
&lt;a name=&quot;top&quot;&gt;

&lt;a href=&quot;#ans&quot;&gt;Jump to Answer&lt;/a&gt;



{{{4x^2+20x-96}}} Start with the given expression.



{{{4(x^2+5x-24)}}} Factor out the GCF {{{4}}}.



Now let&#39;s try to factor the inner expression {{{x^2+5x-24}}}



---------------------------------------------------------------



Looking at the expression {{{x^2+5x-24}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{5}}}, and the last term is {{{-24}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-24}}} to get {{{(1)(-24)=-24}}}.



Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{5}}}?



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 {{{-24}}} (the previous product).



Factors of {{{-24}}}:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-24}}}.

1*(-24)
2*(-12)
3*(-8)
4*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{5}}}:



&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;-24&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-24)=-23&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;-12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-12)=-10&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;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-8)=-5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+(-6)=-2&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;24&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+24=23&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;12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+12=10&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3+8=5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+6=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-3}}} and {{{8}}} add to {{{5}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{8}}} both multiply to {{{-24}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{5}}}



Now replace the middle term {{{5x}}} with {{{-3x+8x}}}. Remember, {{{-3}}} and {{{8}}} add to {{{5}}}. So this shows us that {{{-3x+8x=5x}}}.



{{{x^2+highlight(-3x+8x)-24}}} Replace the second term {{{5x}}} with {{{-3x+8x}}}.



{{{(x^2-3x)+(8x-24)}}} Group the terms into two pairs.



{{{x(x-3)+(8x-24)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-3)+8(x-3)}}} Factor out {{{8}}} 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+8)(x-3)}}} Combine like terms. Or factor out the common term {{{x-3}}}



--------------------------------------------------



So {{{4(x^2+5x-24)}}} then factors further to {{{4(x+8)(x-3)}}}



===============================================================


&lt;a name=&quot;ans&quot;&gt;


Answer:



So {{{4x^2+20x-96}}} completely factors to {{{4(x+8)(x-3)}}}.



Note: you can check the answer by expanding {{{4(x+8)(x-3)}}} to get {{{4x^2+20x-96}}} or by graphing the original expression and the answer (the two graphs should be identical).


&lt;a href=&quot;#top&quot;&gt;Jump to Top&lt;/a&gt;
</PRE>