<PRE><B>Question 729202</B>


Looking at the expression {{{4a^2+7ab-2b^2}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{7}}}, and the last coefficient is {{{-2}}}.



Now multiply the first coefficient {{{4}}} by the last coefficient {{{-2}}} to get {{{(4)(-2)=-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=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;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;/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 {{{7ab}}} with {{{-ab+8ab}}}. Remember, {{{-1}}} and {{{8}}} add to {{{7}}}. So this shows us that {{{-ab+8ab=7ab}}}.



{{{4a^2+highlight(-ab+8ab)-2b^2}}} Replace the second term {{{7ab}}} with {{{-ab+8ab}}}.



{{{(4a^2-ab)+(8ab-2b^2)}}} Group the terms into two pairs.



{{{a(4a-b)+(8ab-2b^2)}}} Factor out the GCF {{{a}}} from the first group.



{{{a(4a-b)+2b(4a-b)}}} Factor out {{{2b}}} 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.



{{{(a+2b)(4a-b)}}} Combine like terms. Or factor out the common term {{{4a-b}}}



===============================================================



Answer:



So {{{4a^2+7ab-2b^2}}} factors to {{{(a+2b)(4a-b)}}}.



In other words, {{{4a^2+7ab-2b^2=(a+2b)(4a-b)}}}.



Note: you can check the answer by expanding {{{(a+2b)(4a-b)}}} to get {{{4a^2+7ab-2b^2}}} or by graphing the original expression and the answer (the two graphs should be identical).
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