<PRE><B>Question 190250</B>


{{{3x^2+27x+42}}} Start with the given expression



{{{3(x^2+9x+14)}}} Factor out the GCF {{{3}}}



Now let&#39;s focus on the inner expression {{{x^2+9x+14}}}





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Looking at the expression {{{x^2+9x+14}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{9}}}, and the last term is {{{14}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{14}}} to get {{{(1)(14)=14}}}.



Now the question is: what two whole numbers multiply to {{{14}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{9}}}?



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 {{{14}}} (the previous product).



Factors of {{{14}}}:

1,2,7,14

-1,-2,-7,-14



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{14}}}.

1*14
2*7
(-1)*(-14)
(-2)*(-7)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{9}}}:



&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;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+14=15&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;2+7=9&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;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-14)=-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;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-7)=-9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{2}}} and {{{7}}} add to {{{9}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{7}}} both multiply to {{{14}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{9}}}



Now replace the middle term {{{9x}}} with {{{2x+7x}}}. Remember, {{{2}}} and {{{7}}} add to {{{9}}}. So this shows us that {{{2x+7x=9x}}}.



{{{x^2+highlight(2x+7x)+14}}} Replace the second term {{{9x}}} with {{{2x+7x}}}.



{{{(x^2+2x)+(7x+14)}}} Group the terms into two pairs.



{{{x(x+2)+(7x+14)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+2)+7(x+2)}}} Factor out {{{7}}} 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+7)(x+2)}}} Combine like terms. Or factor out the common term {{{x+2}}}



So {{{x^2+9x+14}}} factors to {{{(x+7)(x+2)}}}.





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So our expression goes from {{{3(x^2+9x+14)}}} and factors further to {{{3(x+7)(x+2)}}}



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Answer:


So {{{3x^2+27x+42}}} completely factors to {{{3(x+7)(x+2)}}}

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