<PRE><B>Question 206639</B>
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{{{y^2(y+1)-4y(y+1)-21(y+1)}}} Start with the given expression.



{{{(y+1)(y^2-4y-21)}}} Factor out the GCF {{{y+1}}}




Now let&#39;s factor {{{y^2-4y-21}}}:



Looking at the expression {{{y^2-4y-21}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-4}}}, and the last term is {{{-21}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-21}}} to get {{{(1)(-21)=-21}}}.



Now the question is: what two whole numbers multiply to {{{-21}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-4}}}?



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 {{{-21}}} (the previous product).



Factors of {{{-21}}}:

1,3,7,21

-1,-3,-7,-21



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-21}}}.

1*(-21) = -21
3*(-7) = -21
(-1)*(21) = -21
(-3)*(7) = -21


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-4}}}:



&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-21)=-20&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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3+(-7)=-4&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;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+21=20&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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+7=4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{-7}}} add to {{{-4}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{-7}}} both multiply to {{{-21}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-4}}}



Now replace the middle term {{{-4y}}} with {{{3y-7y}}}. Remember, {{{3}}} and {{{-7}}} add to {{{-4}}}. So this shows us that {{{3y-7y=-4y}}}.



{{{y^2+highlight(3y-7y)-21}}} Replace the second term {{{-4y}}} with {{{3y-7y}}}.



{{{(y^2+3y)+(-7y-21)}}} Group the terms into two pairs.



{{{y(y+3)+(-7y-21)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y+3)-7(y+3)}}} 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.



{{{(y-7)(y+3)}}} Combine like terms. Or factor out the common term {{{y+3}}}




So this means that {{{(y+1)(y^2-4y-21)}}} factors to {{{(y+1)(y-7)(y+3)}}}



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Answer:



So {{{y^2(y+1)-4y(y+1)-21(y+1)}}} completely factors to {{{(y+1)(y-7)(y+3)}}}.



In other words, {{{y^2(y+1)-4y(y+1)-21(y+1)=(y+1)(y-7)(y+3)}}}.




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