Question 189830
{{{6a^2-6a-72}}} Start with the given expression



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



Now let's focus on the inner expression {{{a^2-a-12}}}





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


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




Factors of -12:

1,2,3,4,6,12


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


These factors pair up and multiply to -12

(1)*(-12)

(2)*(-6)

(3)*(-4)

(-1)*(12)

(-2)*(6)

(-3)*(4)


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">-12</td><td>1+(-12)=-11</td></tr><tr><td align="center">2</td><td align="center">-6</td><td>2+(-6)=-4</td></tr><tr><td align="center">3</td><td align="center">-4</td><td>3+(-4)=-1</td></tr><tr><td align="center">-1</td><td align="center">12</td><td>-1+12=11</td></tr><tr><td align="center">-2</td><td align="center">6</td><td>-2+6=4</td></tr><tr><td align="center">-3</td><td align="center">4</td><td>-3+4=1</td></tr></table>



From this list we can see that 3 and -4 add up to -1 and multiply to -12



Now looking at the expression {{{a^2-a-12}}}, replace {{{-a}}} with {{{3a-4a}}} (notice {{{3a-4a}}} combines to {{{-a}}}. So it is equivalent to {{{-a}}})


{{{a^2+highlight(3a-4a)-12}}}



Now let's factor {{{a^2+3a-4a-12}}} by grouping:



{{{(a^2+3a)+(-4a-12)}}} Group like terms



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



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


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



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




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So {{{6(a^2-a-12)}}} factors to {{{6(a-4)(a+3)}}}



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Answer: 



So {{{6a^2-6a-72}}} completely factors to {{{6(a-4)(a+3)}}}