Question 182121


{{{12x^2-10x-8}}} Start with the given expression



{{{2(6x^2-5x-4)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{6x^2-5x-4}}}





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Looking at {{{6x^2-5x-4}}} we can see that the first term is {{{6x^2}}} and the last term is {{{-4}}} where the coefficients are 6 and -4 respectively.


Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the  middle coefficient -5? Let's list all of the factors of -24:




Factors of -24:

1,2,3,4,6,8,12,24


-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -24

(1)*(-24)

(2)*(-12)

(3)*(-8)

(4)*(-6)

(-1)*(24)

(-2)*(12)

(-3)*(8)

(-4)*(6)


note: remember, the product of a negative and a positive number is a negative 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">-24</td><td>1+(-24)=-23</td></tr><tr><td align="center">2</td><td align="center">-12</td><td>2+(-12)=-10</td></tr><tr><td align="center"><font color=red>3</font></td><td align="center"><font color=red>-8</font></td><td><font color=red>3+(-8)=-5</font></td></tr><tr><td align="center">4</td><td align="center">-6</td><td>4+(-6)=-2</td></tr><tr><td align="center">-1</td><td align="center">24</td><td>-1+24=23</td></tr><tr><td align="center">-2</td><td align="center">12</td><td>-2+12=10</td></tr><tr><td align="center">-3</td><td align="center">8</td><td>-3+8=5</td></tr><tr><td align="center">-4</td><td align="center">6</td><td>-4+6=2</td></tr></table>



From this list we can see that 3 and -8 add up to -5 and multiply to -24



Now looking at the expression {{{6x^2-5x-4}}}, replace {{{-5x}}} with {{{3x+-8x}}} (notice {{{3x+-8x}}} adds up to {{{-5x}}}. So it is equivalent to {{{-5x}}})


{{{6x^2+highlight(3x+-8x)+-4}}}



Now let's factor {{{6x^2+3x-8x-4}}} by grouping:



{{{(6x^2+3x)+(-8x-4)}}} Group like terms



{{{3x(2x+1)-4(2x+1)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group



{{{(3x-4)(2x+1)}}} Since we have a common term of {{{2x+1}}}, we can combine like terms


So {{{6x^2+3x-8x-4}}} factors to {{{(3x-4)(2x+1)}}}



So this also means that {{{6x^2-5x-4}}} factors to {{{(3x-4)(2x+1)}}} (since {{{6x^2-5x-4}}} is equivalent to {{{6x^2+3x-8x-4}}})




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So our expression goes from {{{2(6x^2-5x-4)}}} and factors further to {{{2(3x-4)(2x+1)}}}



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Answer:


So {{{12x^2-10x-8}}} factors to {{{2(3x-4)(2x+1)}}}