Question 190281


{{{4x^2+16x-48}}} Start with the given expression



{{{4(x^2+4x-12)}}} Factor out the GCF {{{4}}}



Now let's focus on the inner expression {{{x^2+4x-12}}}





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Looking at {{{x^2+4x-12}}} we can see that the first term is {{{x^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 4? 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 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4


<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 -2 and 6 add up to 4 and multiply to -12



Now looking at the expression {{{x^2+4x-12}}}, replace {{{4x}}} with {{{-2x+6x}}} (notice {{{-2x+6x}}} adds up to {{{4x}}}. So it is equivalent to {{{4x}}})


{{{x^2+highlight(-2x+6x)-12}}}



Now let's factor {{{x^2-2x+6x-12}}} by grouping:



{{{(x^2-2x)+(6x-12)}}} Group like terms



{{{x(x-2)+6(x-2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{6}}} out of the second group



{{{(x+6)(x-2)}}} Since we have a common term of {{{x-2}}}, we can combine like terms


So {{{x^2-2x+6x-12}}} factors to {{{(x+6)(x-2)}}}



So this also means that {{{x^2+4x-12}}} factors to {{{(x+6)(x-2)}}} (since {{{x^2+4x-12}}} is equivalent to {{{x^2-2x+6x-12}}})




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So our expression goes from {{{4(x^2+4x-12)}}} and factors further to {{{4(x+6)(x-2)}}}



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Answer:


So {{{4x^2+16x-48}}} completely factors to {{{4(x+6)(x-2)}}}