<PRE><B>Question 807347</B>


{{{3x^2+15x-42}}} Start with the given expression.



{{{3(x^2+5x-14)}}} Factor out the GCF {{{3}}}.



Now let&#39;s try to factor the inner expression {{{x^2+5x-14}}}



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Looking at the expression {{{x^2+5x-14}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{5}}}, 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 {{{5}}}?



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) = -14
2*(-7) = -14
(-1)*(14) = -14
(-2)*(7) = -14


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{5}}}:



&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)=-13&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)=-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;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+14=13&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=5&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 {{{5}}} (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 {{{5}}}



Now replace the middle term {{{5x}}} with {{{-2x+7x}}}. Remember, {{{-2}}} and {{{7}}} add to {{{5}}}. So this shows us that {{{-2x+7x=5x}}}.



{{{x^2+highlight(-2x+7x)-14}}} Replace the second term {{{5x}}} 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}}}



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So {{{3(x^2+5x-14)}}} then factors further to {{{3(x+7)(x-2)}}}



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Answer:



So {{{3x^2+15x-42}}} completely factors to {{{3(x+7)(x-2)}}}.



In other words, {{{3x^2+15x-42=3(x+7)(x-2)}}}.



Note: you can check the answer by expanding {{{3(x+7)(x-2)}}} to get {{{3x^2+15x-42}}} or by graphing the original expression and the answer (the two graphs should be identical).
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