<PRE><B>Question 151747</B>

{{{7x^2+49x+42}}} Start with the given expression



{{{7(x^2+7x+6)}}} Factor out the GCF {{{7}}}



Now let&#39;s focus on the inner expression {{{x^2+7x+6}}}





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Looking at the expression {{{x^2+7x+6}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{7}}}, and the last term is {{{6}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{6}}} to get {{{(1)(6)=6}}}.



Now the question is: what two whole numbers multiply to {{{6}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{7}}}?



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 {{{6}}} (the previous product).



Factors of {{{6}}}:

1,2,3,6

-1,-2,-3,-6



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{6}}}.

1*6
2*3
(-1)*(-6)
(-2)*(-3)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{7}}}:



&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=red&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;1+6=7&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+3=5&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;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-6)=-7&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-3)=-5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{1}}} and {{{6}}} add to {{{7}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{6}}} both multiply to {{{6}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{7}}}



Now replace the middle term {{{7x}}} with {{{x+6x}}}. Remember, {{{1}}} and {{{6}}} add to {{{7}}}. So this shows us that {{{x+6x=7x}}}.



{{{x^2+highlight(x+6x)+6}}} Replace the second term {{{7x}}} with {{{x+6x}}}.



{{{(x^2+x)+(6x+6)}}} Group the terms into two pairs.



{{{x(x+1)+(6x+6)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+1)+6(x+1)}}} Factor out {{{6}}} 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+6)(x+1)}}} Combine like terms. Or factor out the common term {{{x+1}}}






So our expression goes from {{{7(x^2+7x+6)}}} and factors further to {{{7(x+6)(x+1)}}}



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Answer:


So {{{7x^2+49x+42}}} factors to {{{7(x+6)(x+1)}}}

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