Question 139640
Are you sure it's not supposed to read: {{{3x^2 - 3x + 6x - 18}}}???


{{{3x^2 - 3x + 6x - 18}}} Start with the given expression



{{{3x^2 + 3x - 18}}} Combine like terms




{{{3(x^2+x-6)}}} Factor out the GCF {{{3}}}



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





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


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




Factors of -6:

1,2,3,6


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


These factors pair up and multiply to -6

(1)*(-6)

(2)*(-3)

(-1)*(6)

(-2)*(3)


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">-6</td><td>1+(-6)=-5</td></tr><tr><td align="center">2</td><td align="center">-3</td><td>2+(-3)=-1</td></tr><tr><td align="center">-1</td><td align="center">6</td><td>-1+6=5</td></tr><tr><td align="center">-2</td><td align="center">3</td><td>-2+3=1</td></tr></table>



From this list we can see that -2 and 3 add up to 1 and multiply to -6



Now looking at the expression {{{1x^2+1x-6}}}, replace {{{1x}}} with {{{-2x+3x}}} (notice {{{-2x+3x}}} adds up to {{{1x}}}. So it is equivalent to {{{1x}}})


{{{1x^2+highlight(-2x+3x)+-6}}}



Now let's factor {{{1x^2-2x+3x-6}}} by grouping:



{{{(1x^2-2x)+(3x-6)}}} Group like terms



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



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


So {{{1x^2-2x+3x-6}}} factors to {{{(x+3)(x-2)}}}



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




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



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Answer:


So {{{3x^2 - 3x + 6x - 18}}} factors to {{{3(x+3)(x-2)}}}