Question 193101


{{{40x^2-48x-88}}} Start with the given expression



{{{8(5x^2-6x-11)}}} Factor out the GCF {{{8}}}



Now let's focus on the inner expression {{{5x^2-6x-11}}}





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Looking at {{{5x^2-6x-11}}} we can see that the first term is {{{5x^2}}} and the last term is {{{-11}}} where the coefficients are 5 and -11 respectively.


Now multiply the first coefficient 5 and the last coefficient -11 to get -55. Now what two numbers multiply to -55 and add to the  middle coefficient -6? Let's list all of the factors of -55:




Factors of -55:

1,5,11,55


-1,-5,-11,-55 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -55

(1)*(-55)

(5)*(-11)

(-1)*(55)

(-5)*(11)


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



Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-55</td><td>1+(-55)=-54</td></tr><tr><td align="center">5</td><td align="center">-11</td><td>5+(-11)=-6</td></tr><tr><td align="center">-1</td><td align="center">55</td><td>-1+55=54</td></tr><tr><td align="center">-5</td><td align="center">11</td><td>-5+11=6</td></tr></table>



From this list we can see that 5 and -11 add up to -6 and multiply to -55



Now looking at the expression {{{5x^2-6x-11}}}, replace {{{-6x}}} with {{{5x+-11x}}} (notice {{{5x+-11x}}} adds up to {{{-6x}}}. So it is equivalent to {{{-6x}}})


{{{5x^2+highlight(5x+-11x)+-11}}}



Now let's factor {{{5x^2+5x-11x-11}}} by grouping:



{{{(5x^2+5x)+(-11x-11)}}} Group like terms



{{{5x(x+1)-11(x+1)}}} Factor out the GCF of {{{5x}}} out of the first group. Factor out the GCF of {{{-11}}} out of the second group



{{{(5x-11)(x+1)}}} Since we have a common term of {{{x+1}}}, we can combine like terms


So {{{5x^2+5x-11x-11}}} factors to {{{(5x-11)(x+1)}}}



So this also means that {{{5x^2-6x-11}}} factors to {{{(5x-11)(x+1)}}} (since {{{5x^2-6x-11}}} is equivalent to {{{5x^2+5x-11x-11}}})




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So our expression goes from {{{8(5x^2-6x-11)}}} and factors further to {{{8(5x-11)(x+1)}}}



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Answer:


So {{{40x^2-48x-88}}} completely factors to {{{8(5x-11)(x+1)}}}