I'm going to assume you meant to say {{{a^3-2a^2-63a}}}





{{{a^3-2a^2-63a}}} Start with the given expression



{{{a(a^2-2a-63)}}} Factor out the GCF {{{a}}}



Now let's focus on the inner expression {{{a^2-2a-63}}}





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Looking at {{{a^2-2a-63}}} we can see that the first term is {{{a^2}}} and the last term is {{{-63}}} where the coefficients are 1 and -63 respectively.


Now multiply the first coefficient 1 and the last coefficient -63 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 {{{a^2-2a-63}}}, replace {{{-2a}}} with {{{7a-9a}}} (notice {{{7a-9a}}} adds up to {{{-2a}}}. So it is equivalent to {{{-2a}}})


{{{a^2+highlight(7a-9a)-63}}}



Now let's factor {{{a^2+7a-9a-63}}} by grouping:



{{{(a^2+7a)+(-9a-63)}}} Group like terms



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



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


So {{{a^2+7a-9a-63}}} factors to {{{(a-9)(a+7)}}}



So this also means that {{{a^2-2a-63}}} factors to {{{(a-9)(a+7)}}} (since {{{a^2-2a-63}}} is equivalent to {{{a^2+7a-9a-63}}})




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So our expression goes from {{{a(a^2-2a-63)}}} and factors further to {{{a(a-9)(a+7)}}}



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Answer:


So {{{a^3-2a^2-63a}}} completely factors to {{{a(a-9)(a+7)}}}