Question 146173


Looking at {{{x^2-3xy-4y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{-4y^2}}} 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 {{{x^2-3xy-4y^2}}}, replace {{{-3xy}}} with {{{xy+-4xy}}} (notice {{{xy+-4xy}}} adds up to {{{-3xy}}}. So it is equivalent to {{{-3xy}}})


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



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



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



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


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



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




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     Answer:

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




Check:



In order to check our answer, we need to FOIL  {{{(x-4y)(x+y)}}}



Remember, when you FOIL an expression, you follow this procedure:



{{{(highlight(x)-4y)(highlight(x)+y)}}} Multiply the <font color="red">F</font>irst terms:{{{(x)*(x)=x^2}}}.



{{{(highlight(x)-4y)(x+highlight(y))}}} Multiply the <font color="red">O</font>uter terms:{{{(x)*(y)=x*y}}}.



{{{(x+highlight(-4y))(highlight(x)+y)}}} Multiply the <font color="red">I</font>nner terms:{{{(-4*y)*(x)=-4*x*y}}}.



{{{(x+highlight(-4y))(x+highlight(y))}}} Multiply the <font color="red">L</font>ast terms:{{{(-4*y)*(y)=-4*y^2}}}.



{{{x^2+x*y-4*x*y-4*y^2}}} Now collect every term to make a single expression.



{{{x^2-3*x*y-4*y^2}}} Now combine like terms.



So {{{(x-4y)(x+y)}}} FOILS to {{{x^2-3*x*y-4*y^2}}}.



In other words, {{{(x-4y)(x+y)=x^2-3*x*y-4*y^2}}}.



So this verifies our answer.