<PRE><B>Question 177251</B>
# 34




{{{3x^2+6x-24}}} Start with the given expression



{{{3(x^2+2x-8)}}} Factor out the GCF {{{3}}}



Now let&#39;s focus on the inner expression {{{x^2+2x-8}}}





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Looking at {{{x^2+2x-8}}} we can see that the first term is {{{x^2}}} and the last term is {{{-8}}} where the coefficients are 1 and -8 respectively.


Now multiply the first coefficient 1 and the last coefficient -8 to get -8. Now what two numbers multiply to -8 and add to the  middle coefficient 2? Let&#39;s list all of the factors of -8:




Factors of -8:

1,2,4,8


-1,-2,-4,-8 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -8

(1)*(-8)

(2)*(-4)

(-1)*(8)

(-2)*(4)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2


&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;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-8&lt;/td&gt;&lt;td&gt;1+(-8)=-7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-4&lt;/td&gt;&lt;td&gt;2+(-4)=-2&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;8&lt;/td&gt;&lt;td&gt;-1+8=7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;4&lt;/td&gt;&lt;td&gt;-2+4=2&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that -2 and 4 add up to 2 and multiply to -8



Now looking at the expression {{{x^2+2x-8}}}, replace {{{2x}}} with {{{-2x+4x}}} (notice {{{-2x+4x}}} adds up to {{{2x}}}. So it is equivalent to {{{2x}}})


{{{x^2+highlight(-2x+4x)+-8}}}



Now let&#39;s factor {{{x^2-2x+4x-8}}} by grouping:



{{{(x^2-2x)+(4x-8)}}} Group like terms



{{{x(x-2)+4(x-2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{4}}} out of the second group



{{{(x+4)(x-2)}}} Since we have a common term of {{{x-2}}}, we can combine like terms


So {{{x^2-2x+4x-8}}} factors to {{{(x+4)(x-2)}}}



So this also means that {{{x^2+2x-8}}} factors to {{{(x+4)(x-2)}}} (since {{{x^2+2x-8}}} is equivalent to {{{x^2-2x+4x-8}}})




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So our expression goes from {{{3(x^2+2x-8)}}} and factors further to {{{3(x+4)(x-2)}}}



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Answer:


So {{{3x^2+6x-24}}} factors to {{{3(x+4)(x-2)}}}

    



&lt;hr&gt;



# 35





Looking at the expression {{{4x^2+4x-15}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{4}}}, and the last term is {{{-15}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{-15}}} to get {{{(4)(-15)=-60}}}.



Now the question is: what two whole numbers multiply to {{{-60}}} (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 {{{-60}}} (the previous product).



Factors of {{{-60}}}:

1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-60}}}.

1*(-60)
2*(-30)
3*(-20)
4*(-15)
5*(-12)
6*(-10)
(-1)*(60)
(-2)*(30)
(-3)*(20)
(-4)*(15)
(-5)*(12)
(-6)*(10)


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;-60&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-60)=-59&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;-30&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-30)=-28&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;-20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-20)=-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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+(-15)=-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;5&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;5+(-12)=-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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6+(-10)=-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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;60&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+60=59&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;30&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+30=28&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;20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+20=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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+15=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;-5&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;-5+12=7&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-6+10=4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-6}}} and {{{10}}} add to {{{4}}} (the middle coefficient).



So the two numbers {{{-6}}} and {{{10}}} both multiply to {{{-60}}} &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 {{{-6x+10x}}}. Remember, {{{-6}}} and {{{10}}} add to {{{4}}}. So this shows us that {{{-6x+10x=4x}}}.



{{{4x^2+highlight(-6x+10x)-15}}} Replace the second term {{{4x}}} with {{{-6x+10x}}}.



{{{(4x^2-6x)+(10x-15)}}} Group the terms into two pairs.



{{{2x(2x-3)+(10x-15)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(2x-3)+5(2x-3)}}} Factor out {{{5}}} 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.



{{{(2x+5)(2x-3)}}} Combine like terms. Or factor out the common term {{{2x-3}}}


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Answer:



So {{{4x^2+4x-15}}} factors to {{{(2x+5)(2x-3)}}}.



Note: you can check the answer by FOILing {{{(2x+5)(2x-3)}}} to get {{{4x^2+4x-15}}} or by graphing the original expression and the answer (the two graphs should be identical).
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