How do you factor this completely and does it have a GCF? 

x10y3 - 4x9y2 - 21x8y

We can factor out the smallest power of any
letter that appears in all three terms.

x and y both appear in all three terms.

The smallest power of x that appears is x8
The smallest power of y that appears is y1 or just y

So we can factor out x8y as the GCF

x8y(x2y2 - 4xy - 21)

Now we write this:

x8y(xy    )(xy    )

Now we think of two positive integers which
multiply to give -21 and have a sum of -4.

These are -7 and +3

So the completely factored form is:

x8y(xy - 7)(xy + 3)

Edwin