Question 200296
{{{8x^2-16-28x}}} Start with the given expression.



{{{8x^2-28x-16}}} Rearrange the terms.



{{{4(2x^2-7x-4)}}} Factor out the GCF {{{4}}}



Now let's focus on the inner expression {{{2x^2-7x-4}}}





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Looking at {{{2x^2-7x-4}}} we can see that the first term is {{{2x^2}}} and the last term is {{{-4}}} where the coefficients are 2 and -4 respectively.


Now multiply the first coefficient 2 and the last coefficient -4 to get -8. Now what two numbers multiply to -8 and add to the  middle coefficient -7? Let's list all of the factors of -8:




Factors of -8:

1,2,4,8


-1,-2,-4,-8 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -8

(1)*(-8)

(2)*(-4)

(-1)*(8)

(-2)*(4)


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



Now which of these pairs add to -7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -7


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-8</td><td>1+(-8)=-7</td></tr><tr><td align="center">2</td><td align="center">-4</td><td>2+(-4)=-2</td></tr><tr><td align="center">-1</td><td align="center">8</td><td>-1+8=7</td></tr><tr><td align="center">-2</td><td align="center">4</td><td>-2+4=2</td></tr></table>



From this list we can see that 1 and -8 add up to -7 and multiply to -8



Now looking at the expression {{{2x^2-7x-4}}}, replace {{{-7x}}} with {{{1x+-8x}}} (notice {{{1x+-8x}}} adds up to {{{-7x}}}. So it is equivalent to {{{-7x}}})


{{{2x^2+highlight(1x+-8x)+-4}}}



Now let's factor {{{2x^2+1x-8x-4}}} by grouping:



{{{(2x^2+1x)+(-8x-4)}}} Group like terms



{{{x(2x+1)-4(2x+1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group



{{{(x-4)(2x+1)}}} Since we have a common term of {{{2x+1}}}, we can combine like terms


So {{{2x^2+1x-8x-4}}} factors to {{{(x-4)(2x+1)}}}



So this also means that {{{2x^2-7x-4}}} factors to {{{(x-4)(2x+1)}}} (since {{{2x^2-7x-4}}} is equivalent to {{{2x^2+1x-8x-4}}})




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So our expression goes from {{{4(2x^2-7x-4)}}} and factors further to {{{4(x-4)(2x+1)}}}



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Answer:


So {{{8x^2-16-28x}}} completely factors to {{{4(x-4)(2x+1)}}}