Question 127884
Do you mean {{{24x^2+2x-12}}}? Do you want to factor?





{{{24x^2+2x-12}}} Start with the given expression



{{{2(12x^2+x-6)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{12x^2+x-6}}}





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Looking at {{{12x^2+1x-6}}} we can see that the first term is {{{12x^2}}} and the last term is {{{-6}}} where the coefficients are 12 and -6 respectively.


Now multiply the first coefficient 12 and the last coefficient -6 to get -72. Now what two numbers multiply to -72 and add to the  middle coefficient 1? Let's list all of the factors of -72:




Factors of -72:

1,2,3,4,6,8,9,12,18,24,36,72


-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -72

(1)*(-72)

(2)*(-36)

(3)*(-24)

(4)*(-18)

(6)*(-12)

(8)*(-9)

(-1)*(72)

(-2)*(36)

(-3)*(24)

(-4)*(18)

(-6)*(12)

(-8)*(9)


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



Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-72</td><td>1+(-72)=-71</td></tr><tr><td align="center">2</td><td align="center">-36</td><td>2+(-36)=-34</td></tr><tr><td align="center">3</td><td align="center">-24</td><td>3+(-24)=-21</td></tr><tr><td align="center">4</td><td align="center">-18</td><td>4+(-18)=-14</td></tr><tr><td align="center">6</td><td align="center">-12</td><td>6+(-12)=-6</td></tr><tr><td align="center">8</td><td align="center">-9</td><td>8+(-9)=-1</td></tr><tr><td align="center">-1</td><td align="center">72</td><td>-1+72=71</td></tr><tr><td align="center">-2</td><td align="center">36</td><td>-2+36=34</td></tr><tr><td align="center">-3</td><td align="center">24</td><td>-3+24=21</td></tr><tr><td align="center">-4</td><td align="center">18</td><td>-4+18=14</td></tr><tr><td align="center">-6</td><td align="center">12</td><td>-6+12=6</td></tr><tr><td align="center">-8</td><td align="center">9</td><td>-8+9=1</td></tr></table>



From this list we can see that -8 and 9 add up to 1 and multiply to -72



Now looking at the expression {{{12x^2+1x-6}}}, replace {{{1x}}} with {{{-8x+9x}}} (notice {{{-8x+9x}}} adds up to {{{1x}}}. So it is equivalent to {{{1x}}})


{{{12x^2+highlight(-8x+9x)+-6}}}



Now let's factor {{{12x^2-8x+9x-6}}} by grouping:



{{{(12x^2-8x)+(9x-6)}}} Group like terms



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



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


So {{{12x^2-8x+9x-6}}} factors to {{{(4x+3)(3x-2)}}}



So this also means that {{{12x^2+1x-6}}} factors to {{{(4x+3)(3x-2)}}} (since {{{12x^2+1x-6}}} is equivalent to {{{12x^2-8x+9x-6}}})




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So our expression goes from {{{2(12x^2+x-6)}}} and factors further to {{{2(4x+3)(3x-2)}}}



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Answer:


So {{{24x^2+2x-12}}} factors to {{{2(4x+3)(3x-2)}}}