<PRE><B>Question 621898</B>


{{{28x^2+38x-6}}} Start with the given expression.



{{{2(14x^2+19x-3)}}} Factor out the GCF {{{2}}}.



Now let&#39;s try to factor the inner expression {{{14x^2+19x-3}}}



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Looking at the expression {{{14x^2+19x-3}}}, we can see that the first coefficient is {{{14}}}, the second coefficient is {{{19}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{14}}} by the last term {{{-3}}} to get {{{(14)(-3)=-42}}}.



Now the question is: what two whole numbers multiply to {{{-42}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{19}}}?



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 {{{-42}}} (the previous product).



Factors of {{{-42}}}:

1,2,3,6,7,14,21,42

-1,-2,-3,-6,-7,-14,-21,-42



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-42}}}.

1*(-42) = -42
2*(-21) = -42
3*(-14) = -42
6*(-7) = -42
(-1)*(42) = -42
(-2)*(21) = -42
(-3)*(14) = -42
(-6)*(7) = -42


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{19}}}:



&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;-42&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-42)=-41&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-21)=-19&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;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-14)=-11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6+(-7)=-1&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;42&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+42=41&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;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+21=19&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;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+14=11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6+7=1&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{21}}} add to {{{19}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{21}}} both multiply to {{{-42}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{19}}}



Now replace the middle term {{{19x}}} with {{{-2x+21x}}}. Remember, {{{-2}}} and {{{21}}} add to {{{19}}}. So this shows us that {{{-2x+21x=19x}}}.



{{{14x^2+highlight(-2x+21x)-3}}} Replace the second term {{{19x}}} with {{{-2x+21x}}}.



{{{(14x^2-2x)+(21x-3)}}} Group the terms into two pairs.



{{{2x(7x-1)+(21x-3)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(7x-1)+3(7x-1)}}} Factor out {{{3}}} 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.



{{{(2x+3)(7x-1)}}} Combine like terms. Or factor out the common term {{{7x-1}}}



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So {{{2(14x^2+19x-3)}}} then factors further to {{{2(2x+3)(7x-1)}}}



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Answer:



So {{{28x^2+38x-6}}} completely factors to {{{2(2x+3)(7x-1)}}}.



In other words, {{{28x^2+38x-6=2(2x+3)(7x-1)}}}.



Note: you can check the answer by expanding {{{2(2x+3)(7x-1)}}} to get {{{28x^2+38x-6}}} or by graphing the original expression and the answer (the two graphs should be identical).
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