Question 115468
{{{8x^3y^5+16y^3x^5-40x^4y^4}}} Start with the given expression



{{{8x^3y^3(y^2+2x^2-5xy)}}} Factor out the GCF {{{8x^3y^3}}}



{{{8x^3y^3(y^2-5xy+2x^2)}}} Sort the terms


Looking at {{{y^2-5xy+2x^2}}} we can see that the first term is {{{y^2}}} and the last term is {{{2x^2}}} where the coefficients are 1 and 2 respectively.


Now multiply the first coefficient 1 and the last coefficient 2 to get 2. Now what two numbers multiply to 2 and add to the  middle coefficient -5? Let's list all of the factors of 2:




Factors of 2:

1,2


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


These factors pair up and multiply to 2

1*2

(-1)*(-2)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5


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

None of these pairs of factors add to -5. So the inner expression cannot be factored



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Answer:

So that means {{{8x^3y^5+16y^3x^5-40x^4y^4}}} factors to {{{8x^3y^3(y^2+2x^2-5xy)}}}