Question 114364
{{{-3s^2-10s+8}}} Start with the given expression



{{{-(3s^2+10s-8)}}} Factor out the GCF {{{-1}}}



Now let's focus on the inner expression {{{3s^2+10s-8}}}





Looking at {{{3s^2+10s-8}}} we can see that the first term is {{{3s^2}}} and the last term is {{{-8}}} where the coefficients are 3 and -8 respectively.


Now multiply the first coefficient 3 and the last coefficient -8 to get -24. Now what two numbers multiply to -24 and add to the  middle coefficient 10? 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 10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 10


<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 -2 and 12 add up to 10 and multiply to -24



Now looking at the expression {{{3s^2+10s-8}}}, replace {{{10s}}} with {{{-2s+12s}}} (notice {{{-2s+12s}}} adds up to {{{10s}}}. So it is equivalent to {{{10s}}})


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



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



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



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



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


So {{{3s^2-2s+12s-8}}} factors to {{{(s+4)(3s-2)}}}



So this also means that {{{3s^2+10s-8}}} factors to {{{(s+4)(3s-2)}}} (since {{{3s^2+10s-8}}} is equivalent to {{{3s^2-2s+12s-8}}})



{{{-(s+4)(3s-2)}}} Reintroduce the negative

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Answer:


So {{{-3s^2-10s+8}}} factors to {{{-(s+4)(3s-2)}}}