<PRE><B>Question 303540</B>

{{{20r^2+100rg+125g^2}}} Start with the given expression



{{{5(4r^2+20rg+25g^2)}}} Factor out the GCF {{{5}}}



Now let&#39;s focus on the inner expression {{{4r^2+20rg+25g^2}}}





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Looking at {{{4r^2+20rg+25g^2}}} we can see that the first term is {{{4r^2}}} and the last term is {{{25g^2}}} where the coefficients are 4 and 25 respectively.


Now multiply the first coefficient 4 and the last coefficient 25 to get 100. Now what two numbers multiply to 100 and add to the  middle coefficient 20? Let&#39;s list all of the factors of 100:




Factors of 100:

1,2,4,5,10,20,25,50


-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 100

1*100

2*50

4*25

5*20

10*10

(-1)*(-100)

(-2)*(-50)

(-4)*(-25)

(-5)*(-20)

(-10)*(-10)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20


&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;100&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+100=101&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;50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+50=52&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;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+25=29&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;20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+20=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;10&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;10+10=20&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;-100&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-100)=-101&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;-50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-50)=-52&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;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-25)=-29&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;-20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-20)=-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;-10&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;-10+(-10)=-20&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;





From this list we can see that 10 and 10 add up to 20 and multiply to 100



Now looking at the expression {{{4r^2+20rg+25g^2}}}, replace {{{20rg}}} with {{{10rg+10rg}}} (notice {{{10rg+10rg}}} adds up to {{{20rg}}}. So it is equivalent to {{{20rg}}})


{{{4r^2+highlight(10rg+10rg)+25g^2}}}



Now let&#39;s factor {{{4r^2+10rg+10rg+25g^2}}} by grouping:



{{{(4r^2+10rg)+(10rg+25g^2)}}} Group like terms



{{{2r(2r+5g)+5g(2r+5g)}}} Factor out the GCF of {{{2r}}} out of the first group. Factor out the GCF of {{{5g}}} out of the second group



{{{(2r+5g)(2r+5g)}}} Since we have a common term of {{{2r+5g}}}, we can combine like terms


So {{{4r^2+10rg+10rg+25g^2}}} factors to {{{(2r+5g)(2r+5g)}}}



So this also means that {{{4r^2+20rg+25g^2}}} factors to {{{(2r+5g)(2r+5g)}}} (since {{{4r^2+20rg+25g^2}}} is equivalent to {{{4r^2+10rg+10rg+25g^2}}})



note:  {{{(2r+5g)(2r+5g)}}} is equivalent to  {{{(2r+5g)^2}}} since the term {{{2r+5g}}} occurs twice. So {{{4r^2+20rg+25g^2}}} also factors to {{{(2r+5g)^2}}}




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So our expression goes from {{{5(4r^2+20rg+25g^2)}}} and factors further to {{{5(2r+5g)^2}}}



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Answer:


So {{{20r^2+100rg+125g^2}}} factors to {{{5(2r+5g)^2}}}

    

In other words, {{{20r^2+100rg+125g^2=5(2r+5g)^2}}}</PRE>