Question 117350
{{{24x^2+10x-4}}} Start with the given expression



{{{2(12x^2+5x-2)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{12x^2+5x-2}}}





Looking at {{{12x^2+5x-2}}} we can see that the first term is {{{12x^2}}} and the last term is {{{-2}}} where the coefficients are 12 and -2 respectively.


Now multiply the first coefficient 12 and the last coefficient -2 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">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><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 {{{12x^2+5x-2}}}, replace {{{5x}}} with {{{-3x+8x}}} (notice {{{-3x+8x}}} adds up to {{{5x}}}. So it is equivalent to {{{5x}}})


{{{12x^2+highlight(-3x+8x)+-2}}}



Now let's factor {{{12x^2-3x+8x-2}}} by grouping:



{{{(12x^2-3x)+(8x-2)}}} Group like terms



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



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



So {{{12x^2+5x-2}}}factors to {{{(3x+2)(4x-1)}}} 


{{{2(3x+2)(4x-1)}}} Reintroduce the GCF

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Answer:


So {{{24x^2+10x-4}}} factors to {{{2(3x+2)(4x-1)}}}