<PRE><B>Question 299373</B>


Looking at the expression {{{2x^2+15x+25}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{15}}}, and the last term is {{{25}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{25}}} to get {{{(2)(25)=50}}}.



Now the question is: what two whole numbers multiply to {{{50}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{15}}}?



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 {{{50}}} (the previous product).



Factors of {{{50}}}:

1,2,5,10,25,50

-1,-2,-5,-10,-25,-50



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{50}}}.

1*50 = 50
2*25 = 50
5*10 = 50
(-1)*(-50) = 50
(-2)*(-25) = 50
(-5)*(-10) = 50


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{15}}}:



&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;50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+50=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;2&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;2+25=27&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;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5+10=15&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;-50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-50)=-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;-2&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;-2+(-25)=-27&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;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-10)=-15&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{5}}} and {{{10}}} add to {{{15}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{10}}} both multiply to {{{50}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{15}}}



Now replace the middle term {{{15x}}} with {{{5x+10x}}}. Remember, {{{5}}} and {{{10}}} add to {{{15}}}. So this shows us that {{{5x+10x=15x}}}.



{{{2x^2+highlight(5x+10x)+25}}} Replace the second term {{{15x}}} with {{{5x+10x}}}.



{{{(2x^2+5x)+(10x+25)}}} Group the terms into two pairs.



{{{x(2x+5)+(10x+25)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x+5)+5(2x+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)(2x+5)}}} Combine like terms. Or factor out the common term {{{2x+5}}}



===============================================================



Answer:



So {{{2x^2+15x+25}}} factors to {{{(x+5)(2x+5)}}}.



In other words, {{{2x^2+15x+25=(x+5)(2x+5)}}}.



Note: you can check the answer by expanding {{{(x+5)(2x+5)}}} to get {{{2x^2+15x+25}}} or by graphing the original expression and the answer (the two graphs should be identical).
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