Question 219494
I'll do the first one to get you started.



Looking at {{{x^2-4xy-12y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{-12y^2}}} where the coefficients are 1 and -12 respectively.


Now multiply the first coefficient 1 and the last coefficient -12 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">1</td><td align="center">-12</td><td>1+(-12)=-11</td></tr><tr><td align="center"><font color=red>2</font></td><td align="center"><font color=red>-6</font></td><td><font color=red>2+(-6)=-4</font></td></tr><tr><td align="center">3</td><td align="center">-4</td><td>3+(-4)=-1</td></tr><tr><td align="center">-1</td><td align="center">12</td><td>-1+12=11</td></tr><tr><td align="center">-2</td><td align="center">6</td><td>-2+6=4</td></tr><tr><td align="center">-3</td><td align="center">4</td><td>-3+4=1</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 {{{x^2-4xy-12y^2}}}, replace {{{-4xy}}} with {{{2xy-6xy}}} (notice {{{2xy-6xy}}} combines back to {{{-4xy}}}. So it is equivalent to {{{-4xy}}})


{{{x^2+highlight(2xy-6xy)-12y^2}}}



Now let's factor {{{x^2+2xy-6xy-12y^2}}} by grouping:



{{{(x^2+2xy)+(-6xy-12y^2)}}} Group like terms



{{{x(x+2y)-6y(x+2y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-6y}}} out of the second group



{{{(x-6y)(x+2y)}}} Since we have a common term of {{{x+2y}}}, we can combine like terms


So {{{x^2+2xy-6xy-12y^2}}} factors to {{{(x-6y)(x+2y)}}}



So this also means that {{{x^2-4xy-12y^2}}} factors to {{{(x-6y)(x+2y)}}} (since {{{x^2-4xy-12y^2}}} is equivalent to {{{x^2+2xy-6xy-12y^2}}})




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     Answer:

So {{{x^2-4xy-12y^2}}} factors to {{{(x-6y)(x+2y)}}}