Question 182955
Are you sure that the expression is not {{{4x^2-4xy-120y^2}}} ???





{{{4x^2-4xy-120y^2}}} Start with the given expression



{{{4(x^2-xy-30y^2)}}} Factor out the GCF {{{4}}}



Now let's focus on the inner expression {{{x^2-xy-30y^2}}}





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Looking at {{{x^2-xy-30y^2}}} we can see that the first term is {{{1x^2}}} and the last term is {{{-30y^2}}} where the coefficients are 1 and -30 respectively.


Now multiply the first coefficient 1 and the last coefficient -30 to get -30. Now what two numbers multiply to -30 and add to the  middle coefficient -1? Let's list all of the factors of -30:




Factors of -30:

1,2,3,5,6,10,15,30


-1,-2,-3,-5,-6,-10,-15,-30 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -30

(1)*(-30)

(2)*(-15)

(3)*(-10)

(5)*(-6)

(-1)*(30)

(-2)*(15)

(-3)*(10)

(-5)*(6)


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



Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-30</td><td>1+(-30)=-29</td></tr><tr><td align="center">2</td><td align="center">-15</td><td>2+(-15)=-13</td></tr><tr><td align="center">3</td><td align="center">-10</td><td>3+(-10)=-7</td></tr><tr><td align="center">5</td><td align="center">-6</td><td>5+(-6)=-1</td></tr><tr><td align="center">-1</td><td align="center">30</td><td>-1+30=29</td></tr><tr><td align="center">-2</td><td align="center">15</td><td>-2+15=13</td></tr><tr><td align="center">-3</td><td align="center">10</td><td>-3+10=7</td></tr><tr><td align="center">-5</td><td align="center">6</td><td>-5+6=1</td></tr></table>



From this list we can see that 5 and -6 add up to -1 and multiply to -30



Now looking at the expression {{{x^2-xy-30y^2}}}, replace {{{-xy}}} with {{{5xy-6xy}}} (notice {{{5xy-6xy}}} adds up to {{{-xy}}}. So it is equivalent to {{{-xy}}})


{{{x^2+highlight(5xy-6xy)-30y^2}}}



Now let's factor {{{x^2+5xy-6xy-30y^2}}} by grouping:



{{{(x^2+5xy)+(-6xy-30y^2)}}} Group like terms



{{{x(x+5y)-6y(x+5y)}}} 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+5y)}}} Since we have a common term of {{{x+5y}}}, we can combine like terms


So {{{x^2+5xy-6xy-30y^2}}} factors to {{{(x-6y)(x+5y)}}}



So this also means that {{{x^2-xy-30y^2}}} factors to {{{(x-6y)(x+5y)}}} (since {{{x^2-xy-30y^2}}} is equivalent to {{{x^2+5xy-6xy-30y^2}}})




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So our expression goes from {{{4(x^2-xy-30y^2)}}} and factors further to {{{4(x-6y)(x+5y)}}}



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Answer:


So {{{4x^2-4xy-120y^2}}} completely factors to {{{4(x-6y)(x+5y)}}}