Question 186892
I'm assuming that you want to factor.




{{{2x^2-12xy-32y^2}}} Start with the given expression



{{{2(x^2-6xy-16y^2)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{x^2-6xy-16y^2}}}





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Looking at {{{x^2-6xy-16y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{-16y^2}}} where the coefficients are 1 and -16 respectively.


Now multiply the first coefficient 1 and the last coefficient -16 to get -16. Now what two numbers multiply to -16 and add to the  middle coefficient -6? Let's list all of the factors of -16:




Factors of -16:

1,2,4,8


-1,-2,-4,-8 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -16

(1)*(-16)

(2)*(-8)

(-1)*(16)

(-2)*(8)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-16</td><td>1+(-16)=-15</td></tr><tr><td align="center">2</td><td align="center">-8</td><td>2+(-8)=-6</td></tr><tr><td align="center">-1</td><td align="center">16</td><td>-1+16=15</td></tr><tr><td align="center">-2</td><td align="center">8</td><td>-2+8=6</td></tr></table>



From this list we can see that 2 and -8 add up to -6 and multiply to -16



Now looking at the expression {{{x^2-6xy-16y^2}}}, replace {{{-6xy}}} with {{{2xy-8xy}}} (notice {{{2xy-8xy}}} combines to {{{-6xy}}}. So it is equivalent to {{{-6xy}}})



{{{x^2+highlight(2xy-8xy)+-16y^2}}}



Now let's factor {{{1x^2+2xy-8xy-16y^2}}} by grouping:



{{{(x^2+2xy)+(-8xy-16y^2)}}} Group like terms



{{{x(x+2y)-8y(x+2y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-8y}}} out of the second group



{{{(x-8y)(x+2y)}}} Since we have a common term of {{{x+2y}}}, we can combine like terms



So {{{x^2+2xy-8xy-16y^2}}} factors to {{{(x-8y)(x+2y)}}}



So this also means that {{{x^2-6xy-16y^2}}} factors to {{{(x-8y)(x+2y)}}} (since {{{x^2-6xy-16y^2}}} is equivalent to {{{x^2+2xy-8xy-16y^2}}})




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So our expression goes from {{{2(x^2-6xy-16y^2)}}} and factors further to {{{2(x-8y)(x+2y)}}}



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Answer:


So {{{2x^2-12xy-32y^2}}} factors to {{{2(x-8y)(x+2y)}}}