<PRE><B>Question 600707</B>

{{{-36a^2-96ab-64b^2}}} Start with the given expression.



{{{-4(9a^2+24ab+16b^2)}}} Factor out the GCF {{{-4}}}.



Now let&#39;s try to factor the inner expression {{{9a^2+24ab+16b^2}}}



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Looking at the expression {{{9a^2+24ab+16b^2}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{24}}}, and the last coefficient is {{{16}}}.



Now multiply the first coefficient {{{9}}} by the last coefficient {{{16}}} to get {{{(9)(16)=144}}}.



Now the question is: what two whole numbers multiply to {{{144}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{24}}}?



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 {{{144}}} (the previous product).



Factors of {{{144}}}:

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{144}}}.

1*144 = 144
2*72 = 144
3*48 = 144
4*36 = 144
6*24 = 144
8*18 = 144
9*16 = 144
12*12 = 144
(-1)*(-144) = 144
(-2)*(-72) = 144
(-3)*(-48) = 144
(-4)*(-36) = 144
(-6)*(-24) = 144
(-8)*(-18) = 144
(-9)*(-16) = 144
(-12)*(-12) = 144


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{24}}}:



&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;144&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+144=145&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;72&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+72=74&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;48&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+48=51&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;36&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+36=40&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;24&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6+24=30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8+18=26&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9+16=25&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;12+12=24&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;-144&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-144)=-145&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;-72&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-72)=-74&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;-48&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-48)=-51&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;-36&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-36)=-40&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;-24&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6+(-24)=-30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-8+(-18)=-26&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9+(-16)=-25&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-12+(-12)=-24&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{12}}} and {{{12}}} add to {{{24}}} (the middle coefficient).



So the two numbers {{{12}}} and {{{12}}} both multiply to {{{144}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{24}}}



Now replace the middle term {{{24ab}}} with {{{12ab+12ab}}}. Remember, {{{12}}} and {{{12}}} add to {{{24}}}. So this shows us that {{{12ab+12ab=24ab}}}.



{{{9a^2+highlight(12ab+12ab)+16b^2}}} Replace the second term {{{24ab}}} with {{{12ab+12ab}}}.



{{{(9a^2+12ab)+(12ab+16b^2)}}} Group the terms into two pairs.



{{{3a(3a+4b)+(12ab+16b^2)}}} Factor out the GCF {{{3a}}} from the first group.



{{{3a(3a+4b)+4b(3a+4b)}}} Factor out {{{4b}}} 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.



{{{(3a+4b)(3a+4b)}}} Combine like terms. Or factor out the common term {{{3a+4b}}}



{{{(3a+4b)^2}}} Condense the terms.



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So {{{-4(9a^2+24ab+16b^2)}}} then factors further to {{{-4(3a+4b)^2}}}



===============================================================



Answer:



So {{{-36a^2-96ab-64b^2}}} completely factors to {{{-4(3a+4b)^2}}}.



In other words, {{{-36a^2-96ab-64b^2=-4(3a+4b)^2}}}.



Note: you can check the answer by expanding {{{-4(3a+4b)^2}}} to get {{{-36a^2-96ab-64b^2}}} or by graphing the original expression and the answer (the two graphs should be identical).


Check:


{{{-4(3a+4b)^2}}}



{{{-4(3a+4b)(3a+4b)}}}



{{{-4(3a*3a+3a*4b+4b*3a+4b*4b)}}}



{{{-4(9a^2+12ab+12ab+16b^2)}}}



{{{-4(9a^2+24ab+16b^2)}}}



{{{-36a^2-96ab-64b^2}}}



So this verifies the answer.
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