<PRE><B>Question 387339</B>
{{{20b^2+100bt+125t^2}}} Start with the given expression.



{{{5(4b^2+20bt+25t^2)}}} Factor out the GCF {{{5}}}.



Now let&#39;s try to factor the inner expression {{{4b^2+20bt+25t^2}}}



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Looking at the expression {{{4b^2+20bt+25t^2}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{20}}}, and the last coefficient is {{{25}}}.



Now multiply the first coefficient {{{4}}} by the last coefficient {{{25}}} to get {{{(4)(25)=100}}}.



Now the question is: what two whole numbers multiply to {{{100}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{20}}}?



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 {{{100}}} (the previous product).



Factors of {{{100}}}:

1,2,4,5,10,20,25,50,100

-1,-2,-4,-5,-10,-20,-25,-50,-100



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{100}}}.

1*100 = 100
2*50 = 100
4*25 = 100
5*20 = 100
10*10 = 100
(-1)*(-100) = 100
(-2)*(-50) = 100
(-4)*(-25) = 100
(-5)*(-20) = 100
(-10)*(-10) = 100


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{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 the table, we can see that the two numbers {{{10}}} and {{{10}}} add to {{{20}}} (the middle coefficient).



So the two numbers {{{10}}} and {{{10}}} both multiply to {{{100}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{20}}}



Now replace the middle term {{{20bt}}} with {{{10bt+10bt}}}. Remember, {{{10}}} and {{{10}}} add to {{{20}}}. So this shows us that {{{10bt+10bt=20bt}}}.



{{{4b^2+highlight(10bt+10bt)+25t^2}}} Replace the second term {{{20bt}}} with {{{10bt+10bt}}}.



{{{(4b^2+10bt)+(10bt+25t^2)}}} Group the terms into two pairs.



{{{2b(2b+5t)+(10bt+25t^2)}}} Factor out the GCF {{{2b}}} from the first group.



{{{2b(2b+5t)+5t(2b+5t)}}} Factor out {{{5t}}} 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.



{{{(2b+5t)(2b+5t)}}} Combine like terms. Or factor out the common term {{{2b+5t}}}



{{{(2b+5t)^2}}} Condense the terms.



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So {{{5(4b^2+20bt+25t^2)}}} then factors further to {{{5(2b+5t)^2}}}



===============================================================



Answer:



So {{{20b^2+100bt+125t^2}}} completely factors to {{{5(2b+5t)^2}}}.



In other words, {{{20b^2+100bt+125t^2=5(2b+5t)^2}}}.



Note: you can check the answer by expanding {{{5(2b+5t)^2}}} to get {{{20b^2+100bt+125t^2}}} or by graphing the original expression and the answer (the two graphs should be identical).



If you need more help, email me at &lt;a href=&quot;mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help&quot;&gt;jim_thompson5910@hotmail.com&lt;/a&gt;


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Jim</PRE>