Factor completely:

x^10y^3 – 4x^9y^2 – 21x^8y

we can immediately take out an x^8 and a y^1

x^8 y ( x^2 y^2 - 4 x y - 21 )
21 = 3 * 7
3 - 7 = -4
so try
x^8 y ( x y - 7 ) ( x y + 3 )

Start by removing the common factor x^8 y.

x^10y^3 – 4x^9y^2 – 21x^8y
= x^8*y(x^2 y^2 -4 xy -21)
The term in parentheses is a quadratic in the variable xy, and can be factored to

(xy -7)(xy + 3).

Now put it all together

Thanks a lot, this really helps!

To factor the given expression completely, follow these steps:

Step 1: Find the Greatest Common Factor (GCF) of the terms.
In this case, the GCF of the terms is x^8y. Divide each term by x^8y:
x^10y^3 / (x^8y) = x^2y^2
4x^9y^2 / (x^8y) = 4xy
21x^8y / (x^8y) = 21

Step 2: Write the GCF outside the parentheses and divide each term by the GCF.
The factored form is:
x^8y(x^2y^2 - 4xy - 21)

Step 3: Factor the trinomial (x^2y^2 - 4xy - 21).
The trinomial can be factored as (x^2y + 3)(xy - 7).

Therefore, the completely factored form is:
x^8y(x^2y + 3)(xy - 7).