Question 277006

{{{6x^2-8xy-8y^2}}} Start with the given expression



{{{2(3x^2-4xy-4y^2)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{3x^2-4xy-4y^2}}}





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Looking at {{{3x^2-4xy-4y^2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{-4y^2}}} where the coefficients are 3 and -4 respectively.


Now multiply the first coefficient 3 and the last coefficient -4 to get -12. Now what two numbers multiply to -12 and add to the  middle coefficient -4? Let's list all of the factors of -12:




Factors of -12:

1,2,3,4,6,12


-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -12

(1)*(-12)

(2)*(-6)

(3)*(-4)

(-1)*(12)

(-2)*(6)

(-3)*(4)


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



Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>1+(-12)=-11</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>-6</font></td><td  align="center"><font color=red>2+(-6)=-4</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>3+(-4)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-1+12=11</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-2+6=4</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-3+4=1</font></td></tr></table>





From this list we can see that 2 and -6 add up to -4 and multiply to -12



Now looking at the expression {{{3x^2-4xy-4y^2}}}, replace {{{-4xy}}} with {{{2xy-6xy}}} (notice {{{2xy-6xy}}} combines back to {{{-4xy}}}. So it is equivalent to {{{-4xy}}})


{{{3x^2+highlight(2xy-6xy)-4y^2}}}



Now let's factor {{{3x^2+2xy-6xy-4y^2}}} by grouping:



{{{(3x^2+2xy)+(-6xy-4y^2)}}} Group like terms



{{{x(3x+2y)-2y(3x+2y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-2y}}} out of the second group



{{{(x-2y)(3x+2y)}}} Since we have a common term of {{{3x+2y}}}, we can combine like terms


So {{{3x^2+2xy-6xy-4y^2}}} factors to {{{(x-2y)(3x+2y)}}}



So this also means that {{{3x^2-4xy-4y^2}}} factors to {{{(x-2y)(3x+2y)}}} (since {{{3x^2-4xy-4y^2}}} is equivalent to {{{3x^2+2xy-6xy-4y^2}}})




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So our expression goes from {{{2(3x^2-4xy-4y^2)}}} and factors further to {{{2(x-2y)(3x+2y)}}}



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Answer:


So {{{6x^2-8xy-8y^2}}} factors to {{{2(x-2y)(3x+2y)}}}



Note: if you distribute the 2 into the first group, you get {{{(2x-4y)(3x+2y)}}}.