Question 257119


{{{4x^3+8x^2-12x}}} Start with the given expression



{{{4x(x^2+2x-3)}}} Factor out the GCF {{{4x}}}



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



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Looking at {{{x^2+2x-3}}} we can see that the first term is {{{x^2}}} and the last term is {{{-3}}} where the coefficients are 1 and -3 respectively.


Now multiply the first coefficient 1 and the last coefficient -3 to get -3. Now what two numbers multiply to -3 and add to the  middle coefficient 2? Let's list all of the factors of -3:




Factors of -3:

1,3


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


These factors pair up and multiply to -3

(1)*(-3)

(-1)*(3)


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



Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-3</td><td>1+(-3)=-2</td></tr><tr><td align="center">-1</td><td align="center">3</td><td>-1+3=2</td></tr></table>



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



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


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



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



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



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


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



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




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So our expression goes from {{{4x(x^2+2x-3)}}} and factors further to {{{4x(x+3)(x-1)}}}



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Answer:


So {{{4x^3+8x^2-12x}}} factors to {{{4x(x+3)(x-1)}}}