<PRE><B>Question 168159</B>


Looking at {{{3x^2+37xy-86y^2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{-86y^2}}} where the coefficients are 3 and -86 respectively.


Now multiply the first coefficient 3 and the last coefficient -86 to get -258. Now what two numbers multiply to -258 and add to the  middle coefficient 37? Let&#39;s list all of the factors of -258:




Factors of -258:

1,2,3,6,43,86,129,258


-1,-2,-3,-6,-43,-86,-129,-258 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -258

(1)*(-258)

(2)*(-129)

(3)*(-86)

(6)*(-43)

(-1)*(258)

(-2)*(129)

(-3)*(86)

(-6)*(43)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 37? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 37


&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;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-258&lt;/td&gt;&lt;td&gt;1+(-258)=-257&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-129&lt;/td&gt;&lt;td&gt;2+(-129)=-127&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-86&lt;/td&gt;&lt;td&gt;3+(-86)=-83&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;6&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-43&lt;/td&gt;&lt;td&gt;6+(-43)=-37&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;258&lt;/td&gt;&lt;td&gt;-1+258=257&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;129&lt;/td&gt;&lt;td&gt;-2+129=127&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;86&lt;/td&gt;&lt;td&gt;-3+86=83&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-6&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;43&lt;/td&gt;&lt;td&gt;-6+43=37&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that -6 and 43 add up to 37 and multiply to -258



Now looking at the expression {{{3x^2+37xy-86y^2}}}, replace {{{37xy}}} with {{{-6xy+43xy}}} (notice {{{-6xy+43xy}}} adds up to {{{37xy}}}. So it is equivalent to {{{37xy}}})


{{{3x^2+highlight(-6xy+43xy)+-86y^2}}}



Now let&#39;s factor {{{3x^2-6xy+43xy-86y^2}}} by grouping:



{{{(3x^2-6xy)+(43xy-86y^2)}}} Group like terms



{{{3x(x-2y)+43y(x-2y)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{43y}}} out of the second group



{{{(3x+43y)(x-2y)}}} Since we have a common term of {{{x-2y}}}, we can combine like terms


So {{{3x^2-6xy+43xy-86y^2}}} factors to {{{(3x+43y)(x-2y)}}}



So this also means that {{{3x^2+37xy-86y^2}}} factors to {{{(3x+43y)(x-2y)}}} (since {{{3x^2+37xy-86y^2}}} is equivalent to {{{3x^2-6xy+43xy-86y^2}}})



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Answer:


So {{{3x^2+37xy-86y^2}}} factors to {{{(3x+43y)(x-2y)}}} 

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