<PRE><B>Question 190254</B>


Looking at the expression {{{x^2+18x+45}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{18}}}, and the last term is {{{45}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{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 {{{18}}}?



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
3*15
5*9
(-1)*(-45)
(-3)*(-15)
(-5)*(-9)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{18}}}:



&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=red&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&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;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 {{{3}}} and {{{15}}} add to {{{18}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{15}}} both multiply to {{{45}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{18}}}



Now replace the middle term {{{18x}}} with {{{3x+15x}}}. Remember, {{{3}}} and {{{15}}} add to {{{18}}}. So this shows us that {{{3x+15x=18x}}}.



{{{x^2+highlight(3x+15x)+45}}} Replace the second term {{{18x}}} with {{{3x+15x}}}.



{{{(x^2+3x)+(15x+45)}}} Group the terms into two pairs.



{{{x(x+3)+(15x+45)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+3)+15(x+3)}}} Factor out {{{15}}} 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+15)(x+3)}}} Combine like terms. Or factor out the common term {{{x+3}}}


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Answer:



So {{{x^2+18x+45}}} factors to {{{(x+15)(x+3)}}}.



Note: you can check the answer by FOILing {{{(x+15)(x+3)}}} to get {{{x^2+18x+45}}} or by graphing the original expression and the answer (the two graphs should be identical).
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