<PRE><B>Question 182948</B>


{{{4x^2+40x+100}}} Start with the given expression



{{{4(x^2+10x+25)}}} Factor out the GCF {{{4}}}



Now let&#39;s focus on the inner expression {{{x^2+10x+25}}}





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Looking at the expression {{{x^2+10x+25}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{10}}}, and the last term is {{{25}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{25}}} to get {{{(1)(25)=25}}}.



Now the question is: what two whole numbers multiply to {{{25}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{10}}}?



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 {{{25}}} (the previous product).



Factors of {{{25}}}:

1,5,25

-1,-5,-25



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{25}}}.

1*25
5*5
(-1)*(-25)
(-5)*(-5)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{10}}}:



&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;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+25=26&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5+5=10&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;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-25)=-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;-5&lt;/font&gt;&lt;/td&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;-5+(-5)=-10&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{5}}} and {{{5}}} add to {{{10}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{5}}} both multiply to {{{25}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{10}}}



Now replace the middle term {{{10x}}} with {{{5x+5x}}}. Remember, {{{5}}} and {{{5}}} add to {{{10}}}. So this shows us that {{{5x+5x=10x}}}.



{{{x^2+highlight(5x+5x)+25}}} Replace the second term {{{10x}}} with {{{5x+5x}}}.



{{{(x^2+5x)+(5x+25)}}} Group the terms into two pairs.



{{{x(x+5)+(5x+25)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+5)+5(x+5)}}} 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.



{{{(x+5)(x+5)}}} Combine like terms. Or factor out the common term {{{x+5}}}



{{{(x+5)^2}}} Condense the terms.



So {{{x^2+10x+25}}} factors to {{{(x+5)^2}}}.





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So our expression goes from {{{4(x^2+10x+25)}}} and factors further to {{{4(x+5)^2}}}



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Answer:


So {{{4x^2+40x+100}}} factors to {{{4(x+5)^2}}}

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