Question 125429
Let {{{z=x-2y}}}



So the expression {{{3(x-2y)^2 + 6(x-2y) - 45 }}} becomes {{{3z^2 + 6z - 45}}} 





{{{3z^2+6z-45}}} Start with the given expression



{{{3(z^2+2z-15)}}} Factor out the GCF {{{3}}}



Now let's focus on the inner expression {{{z^2+2z-15}}}





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Looking at {{{1z^2+2z-15}}} we can see that the first term is {{{1z^2}}} and the last term is {{{-15}}} where the coefficients are 1 and -15 respectively.


Now multiply the first coefficient 1 and the last coefficient -15 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient 2? Let's list all of the factors of -15:




Factors of -15:

1,3,5,15


-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -15

(1)*(-15)

(3)*(-5)

(-1)*(15)

(-3)*(5)


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



Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-15</td><td>1+(-15)=-14</td></tr><tr><td align="center">3</td><td align="center">-5</td><td>3+(-5)=-2</td></tr><tr><td align="center">-1</td><td align="center">15</td><td>-1+15=14</td></tr><tr><td align="center">-3</td><td align="center">5</td><td>-3+5=2</td></tr></table>



From this list we can see that -3 and 5 add up to 2 and multiply to -15



Now looking at the expression {{{1z^2+2z-15}}}, replace {{{2z}}} with {{{-3z+5z}}} (notice {{{-3z+5z}}} adds up to {{{2z}}}. So it is equivalent to {{{2z}}})


{{{1z^2+highlight(-3z+5z)+-15}}}



Now let's factor {{{1z^2-3z+5z-15}}} by grouping:



{{{(1z^2-3z)+(5z-15)}}} Group like terms



{{{z(z-3)+5(z-3)}}} Factor out the GCF of {{{z}}} out of the first group. Factor out the GCF of {{{5}}} out of the second group



{{{(z+5)(z-3)}}} Since we have a common term of {{{z-3}}}, we can combine like terms


So {{{1z^2-3z+5z-15}}} factors to {{{(z+5)(z-3)}}}



So this also means that {{{1z^2+2z-15}}} factors to {{{(z+5)(z-3)}}} (since {{{1z^2+2z-15}}} is equivalent to {{{1z^2-3z+5z-15}}})




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So our expression goes from {{{3(z^2+2z-15)}}} and factors further to {{{3(z+5)(z-3)}}}





Remember we let {{{z=x-2y}}}, so let's replace z with x-2y


{{{3(x-2y+5)(x-2y-3)}}} Plug in {{{z=x-2y}}}





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Answer:


So {{{3(x-2y)^2 + 6(x-2y) - 45 }}} factors to {{{3(x-2y+5)(x-2y-3)}}}



So the answer is D)