Question 101511
Given:
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{{{x^10y^3 - 4x^9y^2 - 21x^8y}}}
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First look at the numbers that multiply each of the terms to see if there is are common
factors. The first term is multiplied by 1 (understood), the second term by -4, and the
third term by -21.  These three numbers have no common factors, so we don't change them.
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Next look at the x part of each of the terms. The first term contains {{{x^10}}}, the second
term {{{x^9}}}, and the third term {{{x^8}}}. {{{x^8}}} is common to each term because
{{{x^10 = x^2*x^8}}} and {{{x^9 = x*x^8}}}. So we can pull an {{{x^8}}} from each term and
we get:

{{{x^10y^3 - 4x^9y^2 - 21x^8y = (x^8)*(x^2*y^3 - 4x*y^2 -21y)}}}
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Next notice that you have a "y" common to all terms. So pull a y out as a multiplier and 
you have:
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{{{(x^8)*(x^2*y^3 - 4x*y^2 -21y) = (x^8)*(y)(x^2y^2 -4xy - 21)}}}
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Notice that we can multiply out {{{(x^8)*(y)(x^2y^2 -4xy-21)}}} and the result should be
the original given expression. This is just a "check".
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So we are down to {{{(x^8)*(y)(x^2y^2 -4xy-21)}}}. We now need to see if we can factor the
{{{x^2y^2 - 4xy -21}}} Yes we can.  It might be a little easier to see if we wrote 
{{{x^2y^2 }}} in an equivalent form of {{{(xy)^2}}} to change the expression to:
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{{{(xy)^2 - 4xy - 21}}} and we might see it a little more clearly if we let xy = A to get:
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{{{A^2 - 4A - 21}}}
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We can try to factor this into (A + ___)*(A + ___). The underscores have to be two numbers
that are factors of -21. These numbers could be 21 and 1 or 7 and 3. With their appropriate signs
they must sum to -4.  -7 and + 3 do that. If they are multiplied they give -21 and if they are
added they give -4 which is the multiplier of the center term that contains just A.
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So we can write that {{{A^2 - 4A - 21}}} factors to {{{(A - 7)*(A+ 3)}}}. But also recall
that we said A was equal to xy. So now we can substitute xy into the two factors and get:
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{{{(A-7)*(A+3) = (xy - 7)*(xy + 3)}}}
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And so we can say that {{{x^2y^2 - 4xy - 21}}} factors to {{{(xy - 7)*(xy+3)}}} and we can 
carry this result back into this equation that we got earlier:
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{{{(x^8)*(y)(x^2y^2 -4xy-21)}}}
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In this equation, replace {{{(x^2y^2 -4xy-21)}}} with {{{(xy - 7)*(xy+3)}}} and the result is:
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{{{(x^8)*(y)*(xy - 7)*(xy+3)}}}
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and that's the answer to your problem.
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I know this is confusing, but I hope that you can see your way through it. If you can
understand all the maneuvering in this you will have made a lot of progress.