Question 174855


{{{6s^2+40s-64}}} Start with the given expression



{{{2(3s^2+20s-32)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{3s^2+20s-32}}}





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Looking at {{{3s^2+20s-32}}} we can see that the first term is {{{3s^2}}} and the last term is {{{-32}}} where the coefficients are 3 and -32 respectively.


Now multiply the first coefficient 3 and the last coefficient -32 to get -96. Now what two numbers multiply to -96 and add to the  middle coefficient 20? Let's list all of the factors of -96:




Factors of -96:

1,2,3,4,6,8,12,16,24,32,48,96


-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -96

(1)*(-96)

(2)*(-48)

(3)*(-32)

(4)*(-24)

(6)*(-16)

(8)*(-12)

(-1)*(96)

(-2)*(48)

(-3)*(32)

(-4)*(24)

(-6)*(16)

(-8)*(12)


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



Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-96</td><td>1+(-96)=-95</td></tr><tr><td align="center">2</td><td align="center">-48</td><td>2+(-48)=-46</td></tr><tr><td align="center">3</td><td align="center">-32</td><td>3+(-32)=-29</td></tr><tr><td align="center">4</td><td align="center">-24</td><td>4+(-24)=-20</td></tr><tr><td align="center">6</td><td align="center">-16</td><td>6+(-16)=-10</td></tr><tr><td align="center">8</td><td align="center">-12</td><td>8+(-12)=-4</td></tr><tr><td align="center">-1</td><td align="center">96</td><td>-1+96=95</td></tr><tr><td align="center">-2</td><td align="center">48</td><td>-2+48=46</td></tr><tr><td align="center">-3</td><td align="center">32</td><td>-3+32=29</td></tr><tr><td align="center">-4</td><td align="center">24</td><td>-4+24=20</td></tr><tr><td align="center">-6</td><td align="center">16</td><td>-6+16=10</td></tr><tr><td align="center">-8</td><td align="center">12</td><td>-8+12=4</td></tr></table>



From this list we can see that -4 and 24 add up to 20 and multiply to -96



Now looking at the expression {{{3s^2+20s-32}}}, replace {{{20s}}} with {{{-4s+24s}}} (notice {{{-4s+24s}}} adds up to {{{20s}}}. So it is equivalent to {{{20s}}})


{{{3s^2+highlight(-4s+24s)+-32}}}



Now let's factor {{{3s^2-4s+24s-32}}} by grouping:



{{{(3s^2-4s)+(24s-32)}}} Group like terms



{{{s(3s-4)+8(3s-4)}}} Factor out the GCF of {{{s}}} out of the first group. Factor out the GCF of {{{8}}} out of the second group



{{{(s+8)(3s-4)}}} Since we have a common term of {{{3s-4}}}, we can combine like terms


So {{{3s^2-4s+24s-32}}} factors to {{{(s+8)(3s-4)}}}



So this also means that {{{3s^2+20s-32}}} factors to {{{(s+8)(3s-4)}}} (since {{{3s^2+20s-32}}} is equivalent to {{{3s^2-4s+24s-32}}})




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So our expression goes from {{{2(3s^2+20s-32)}}} and factors further to {{{2(s+8)(3s-4)}}}



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Answer:


So {{{6s^2+40s-64}}} factors to {{{2(s+8)(3s-4)}}}

    

So the two binomial factors are {{{s+8}}} and {{{3s-4}}}