<PRE><B>Question 695305</B>


Looking at the expression {{{x^2+14xy+45y^2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{14}}}, and the last coefficient is {{{45}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{45}}} to get {{{(1)(45)=45}}}.



Now the question is: what two whole numbers multiply to {{{45}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{14}}}?



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 {{{45}}} (the previous product).



Factors of {{{45}}}:

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{45}}}.

1*45 = 45
3*15 = 45
5*9 = 45
(-1)*(-45) = 45
(-3)*(-15) = 45
(-5)*(-9) = 45


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{14}}}:



&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;45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+45=46&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;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+15=18&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5+9=14&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;-45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-45)=-46&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;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-15)=-18&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-9)=-14&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{5}}} and {{{9}}} add to {{{14}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{9}}} both multiply to {{{45}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{14}}}



Now replace the middle term {{{14xy}}} with {{{5xy+9xy}}}. Remember, {{{5}}} and {{{9}}} add to {{{14}}}. So this shows us that {{{5xy+9xy=14xy}}}.



{{{x^2+highlight(5xy+9xy)+45y^2}}} Replace the second term {{{14xy}}} with {{{5xy+9xy}}}.



{{{(x^2+5xy)+(9xy+45y^2)}}} Group the terms into two pairs.



{{{x(x+5y)+(9xy+45y^2)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+5y)+9y(x+5y)}}} Factor out {{{9y}}} 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+9y)(x+5y)}}} Combine like terms. Or factor out the common term {{{x+5y}}}



===============================================================



Answer:



So {{{x^2+14xy+45y^2}}} factors to {{{(x+9y)(x+5y)}}}.



In other words, {{{x^2+14xy+45y^2=(x+9y)(x+5y)}}}.



Note: you can check the answer by expanding {{{(x+9y)(x+5y)}}} to get {{{x^2+14xy+45y^2}}} or by graphing the original expression and the answer (the two graphs should be identical).
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