Question 236686
I'll help you with the first two to get you going.


# 1


Looking at {{{9x^2-35xy-4y^2}}} we can see that the first term is {{{9x^2}}} and the last term is {{{-4y^2}}} where the coefficients are 9 and -4 respectively.


Now multiply the first coefficient 9 and the last coefficient -4 to get -36. Now what two numbers multiply to -36 and add to the  middle coefficient -35? Let's list all of the factors of -36:




Factors of -36:

1,2,3,4,6,9,12,18


-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -36

(1)*(-36)

(2)*(-18)

(3)*(-12)

(4)*(-9)

(-1)*(36)

(-2)*(18)

(-3)*(12)

(-4)*(9)


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



Now which of these pairs add to -35? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -35


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-36</td><td>1+(-36)=-35</td></tr><tr><td align="center">2</td><td align="center">-18</td><td>2+(-18)=-16</td></tr><tr><td align="center">3</td><td align="center">-12</td><td>3+(-12)=-9</td></tr><tr><td align="center">4</td><td align="center">-9</td><td>4+(-9)=-5</td></tr><tr><td align="center">-1</td><td align="center">36</td><td>-1+36=35</td></tr><tr><td align="center">-2</td><td align="center">18</td><td>-2+18=16</td></tr><tr><td align="center">-3</td><td align="center">12</td><td>-3+12=9</td></tr><tr><td align="center">-4</td><td align="center">9</td><td>-4+9=5</td></tr></table>



From this list we can see that 1 and -36 add up to -35 and multiply to -36



Now looking at the expression {{{9x^2-35xy-4y^2}}}, replace {{{-35xy}}} with {{{1xy+-36xy}}} (notice {{{1xy+-36xy}}} adds up to {{{-35xy}}}. So it is equivalent to {{{-35xy}}})


{{{9x^2+highlight(1xy+-36xy)+-4y^2}}}



Now let's factor {{{9x^2+1xy-36xy-4y^2}}} by grouping:



{{{(9x^2+1xy)+(-36xy-4y^2)}}} Group like terms



{{{x(9x+y)-4y(9x+y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-4y}}} out of the second group



{{{(x-4y)(9x+y)}}} Since we have a common term of {{{9x+y}}}, we can combine like terms


So {{{9x^2+1xy-36xy-4y^2}}} factors to {{{(x-4y)(9x+y)}}}



So this also means that {{{9x^2-35xy-4y^2}}} factors to {{{(x-4y)(9x+y)}}} (since {{{9x^2-35xy-4y^2}}} is equivalent to {{{9x^2+1xy-36xy-4y^2}}})




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     Answer:

So {{{9x^2-35xy-4y^2}}} completely factors to {{{(x-4y)(9x+y)}}}


In other words, {{{9x^2-35xy-4y^2=(x-4y)(9x+y)}}}



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# 2



{{{2x^2-xy+10xy-5y^2}}} Start with the given expression



{{{(2x^2-xy)+(10xy-5y^2)}}} Group like terms



{{{x(2x-y)+5y(2x-y)}}} Factor out the GCF {{{x}}} out of the first group. Factor out the GCF {{{5y}}} out of the second group



{{{(x+5y)(2x-y)}}} Since we have the common term {{{2x-y}}}, we can combine like terms



So {{{2x^2-xy+10xy-5y^2}}} factors to {{{(x+5y)(2x-y)}}}



In other words, {{{2x^2-xy+10xy-5y^2=(x+5y)(2x-y)}}}