Question 115729
Do you want to factor?


{{{x^2y^2+3x^2y-4x^2}}} Start with the given expression



{{{x^2(y^2+3y-4)}}} Factor out the GCF {{{x^2}}}



Now let's focus on the inner expression {{{y^2+3y-4}}}





Looking at {{{y^2+3y-4}}} we can see that the first term is {{{y^2}}} and the last term is {{{-4}}} where the coefficients are 1 and -4 respectively.


Now multiply the first coefficient 1 and the last coefficient -4 to get -4. Now what two numbers multiply to -4 and add to the  middle coefficient 3? Let's list all of the factors of -4:




Factors of -4:

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 -4

(1)*(-4)

(-1)*(4)


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



Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-4</td><td>1+(-4)=-3</td></tr><tr><td align="center">-1</td><td align="center">4</td><td>-1+4=3</td></tr></table>



From this list we can see that -1 and 4 add up to 3 and multiply to -4



Now looking at the expression {{{y^2+3y-4}}}, replace {{{3y}}} with {{{-1y+4y}}} (notice {{{-1y+4y}}} adds up to {{{3y}}}. So it is equivalent to {{{3y}}})


{{{y^2+highlight(-1y+4y)+-4}}}



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



{{{(y^2-1y)+(4y-4)}}} Group like terms



{{{y(y-1)+4(y-1)}}} Factor out the GCF of {{{y}}} out of the first group. Factor out the GCF of {{{4}}} out of the second group



{{{(y+4)(y-1)}}} Since we have a common term of {{{y-1}}}, we can combine like terms


So {{{y^2-1y+4y-4}}} factors to {{{(y+4)(y-1)}}}



So this also means that {{{y^2+3y-4}}} factors to {{{(y+4)(y-1)}}} (since {{{y^2+3y-4}}} is equivalent to {{{y^2-1y+4y-4}}})



{{{x^2(y+4)(y-1)}}} Now reintroduce the GCF


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Answer:


So {{{x^2y^2+3x^2y-4x^2}}} factors to {{{x^2(y+4)(y-1)}}}