Question 145714
I'm assuming you want to factor right?





{{{36x^4-8x^3-28x^2}}} Start with the given expression



{{{4x^2(9x^2-2x-7)}}} Factor out the GCF {{{4x^2}}}



Now let's focus on the inner expression {{{9x^2-2x-7}}}





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Looking at {{{9x^2-2x-7}}} we can see that the first term is {{{9x^2}}} and the last term is {{{-7}}} where the coefficients are 9 and -7 respectively.


Now multiply the first coefficient 9 and the last coefficient -7 to get -63. Now what two numbers multiply to -63 and add to the  middle coefficient -2? Let's list all of the factors of -63:




Factors of -63:

1,3,7,9,21,63


-1,-3,-7,-9,-21,-63 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -63

(1)*(-63)

(3)*(-21)

(7)*(-9)

(-1)*(63)

(-3)*(21)

(-7)*(9)


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">-63</td><td>1+(-63)=-62</td></tr><tr><td align="center">3</td><td align="center">-21</td><td>3+(-21)=-18</td></tr><tr><td align="center">7</td><td align="center">-9</td><td>7+(-9)=-2</td></tr><tr><td align="center">-1</td><td align="center">63</td><td>-1+63=62</td></tr><tr><td align="center">-3</td><td align="center">21</td><td>-3+21=18</td></tr><tr><td align="center">-7</td><td align="center">9</td><td>-7+9=2</td></tr></table>



From this list we can see that 7 and -9 add up to -2 and multiply to -63



Now looking at the expression {{{9x^2-2x-7}}}, replace {{{-2x}}} with {{{7x+-9x}}} (notice {{{7x+-9x}}} adds up to {{{-2x}}}. So it is equivalent to {{{-2x}}})


{{{9x^2+highlight(7x+-9x)+-7}}}



Now let's factor {{{9x^2+7x-9x-7}}} by grouping:



{{{(9x^2+7x)+(-9x-7)}}} Group like terms



{{{x(9x+7)-1(9x+7)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-1}}} out of the second group



{{{(x-1)(9x+7)}}} Since we have a common term of {{{9x+7}}}, we can combine like terms


So {{{9x^2+7x-9x-7}}} factors to {{{(x-1)(9x+7)}}}



So this also means that {{{9x^2-2x-7}}} factors to {{{(x-1)(9x+7)}}} (since {{{9x^2-2x-7}}} is equivalent to {{{9x^2+7x-9x-7}}})




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So our expression goes from {{{4x^2(9x^2-2x-7)}}} and factors further to {{{4x^2(x-1)(9x+7)}}}



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Answer:


So {{{36x^4-8x^3-28x^2}}} factors to {{{4x^2(x-1)(9x+7)}}}