<PRE><B>Question 207781</B>
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{{{120p+100p^2+36}}}  Start with the given expression.



{{{100p^2+120p+36}}} Rearrange the terms.



{{{4(25p^2+30p+9)}}} Factor out the GCF {{{4}}}.



Now let&#39;s try to factor the inner expression {{{25p^2+30p+9}}}



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Looking at the expression {{{25p^2+30p+9}}}, we can see that the first coefficient is {{{25}}}, the second coefficient is {{{30}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{25}}} by the last term {{{9}}} to get {{{(25)(9)=225}}}.



Now the question is: what two whole numbers multiply to {{{225}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{30}}}?



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 {{{225}}} (the previous product).



Factors of {{{225}}}:

1,3,5,9,15,25,45,75,225

-1,-3,-5,-9,-15,-25,-45,-75,-225



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{225}}}.

1*225 = 225
3*75 = 225
5*45 = 225
9*25 = 225
15*15 = 225
(-1)*(-225) = 225
(-3)*(-75) = 225
(-5)*(-45) = 225
(-9)*(-25) = 225
(-15)*(-15) = 225


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{30}}}:



&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;225&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+225=226&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;75&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+75=78&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;45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+45=50&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;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9+25=34&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15+15=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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-225&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-225)=-226&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;-75&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-75)=-78&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;-45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-45)=-50&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;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9+(-25)=-34&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-15+(-15)=-30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{15}}} and {{{15}}} add to {{{30}}} (the middle coefficient).



So the two numbers {{{15}}} and {{{15}}} both multiply to {{{225}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{30}}}



Now replace the middle term {{{30p}}} with {{{15p+15p}}}. Remember, {{{15}}} and {{{15}}} add to {{{30}}}. So this shows us that {{{15p+15p=30p}}}.



{{{25p^2+highlight(15p+15p)+9}}} Replace the second term {{{30p}}} with {{{15p+15p}}}.



{{{(25p^2+15p)+(15p+9)}}} Group the terms into two pairs.



{{{5p(5p+3)+(15p+9)}}} Factor out the GCF {{{5p}}} from the first group.



{{{5p(5p+3)+3(5p+3)}}} Factor out {{{3}}} 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.



{{{(5p+3)(5p+3)}}} Combine like terms. Or factor out the common term {{{5p+3}}}



{{{(5p+3)^2}}} Condense the terms.



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So {{{4(25p^2+30p+9)}}} then factors further to {{{4(5p+3)^2}}}



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Answer:



So {{{120p+100p^2+36}}} completely factors to {{{4(5p+3)^2}}}.



In other words, {{{120p+100p^2+36=4(5p+3)^2}}}.



Note: you can check the answer by expanding {{{4(5p+3)^2}}} to get {{{120p+100p^2+36}}} or by graphing the original expression and the answer (the two graphs should be identical).


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