<PRE><B>Question 778022</B>


Looking at the expression {{{2x^2-7x-4}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-7}}}, and the last term is {{{-4}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-4}}} to get {{{(2)(-4)=-8}}}.



Now the question is: what two whole numbers multiply to {{{-8}}} (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 {{{-8}}} (the previous product).



Factors of {{{-8}}}:

1,2,4,8

-1,-2,-4,-8



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-8}}}.

1*(-8) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8


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;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;1+(-8)=-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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-4)=-2&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;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+8=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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+4=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{1}}} and {{{-8}}} add to {{{-7}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-8}}} both multiply to {{{-8}}} &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-8x}}}. Remember, {{{1}}} and {{{-8}}} add to {{{-7}}}. So this shows us that {{{x-8x=-7x}}}.



{{{2x^2+highlight(x-8x)-4}}} Replace the second term {{{-7x}}} with {{{x-8x}}}.



{{{(2x^2+x)+(-8x-4)}}} Group the terms into two pairs.



{{{x(2x+1)+(-8x-4)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x+1)-4(2x+1)}}} Factor out {{{4}}} 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-4)(2x+1)}}} Combine like terms. Or factor out the common term {{{2x+1}}}



===============================================================



Answer:



So {{{2x^2-7x-4}}} factors to {{{(x-4)(2x+1)}}}.



In other words, {{{2x^2-7x-4=(x-4)(2x+1)}}}.



Note: you can check the answer by expanding {{{(x-4)(2x+1)}}} to get {{{2x^2-7x-4}}} or by graphing the original expression and the answer (the two graphs should be identical).
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