To find the degree of the given polynomial, let's analyze each term.

The given polynomial is:

$-4u + 15w^9 u^7 v^4 - v^8 w^9 - 6$

1. Identify the degree of each term:

   • First term: $-4u$ — The degree of $u$ is 1.
   • Second term: $15w^9 u^7 v^4$ — The degree is the sum of the powers of $w$, $u$, and $v$, which is $9 + 7 + 4 = 20$.
   • Third term: $-v^8 w^9$ — The degree is the sum of the powers of $v$ and $w$, which is $8 + 9 = 17$.
   • Fourth term: $-6$ — This is a constant term, so its degree is 0.

2. Find the highest degree term: The second term, $15w^9 u^7 v^4$, has the highest degree, which is 20.

Thus, the degree of the polynomial is 20.

Would you like more details on how I broke down the polynomial, or do you have any other questions?

Here are 5 related questions to expand on this:

1. How do you determine the degree of a polynomial with multiple variables?
2. What is the degree of a monomial like $x^3 y^2 z^5$?
3. How does the degree of a polynomial affect its graph?
4. Can a polynomial have negative exponents, and how would that affect its degree?
5. What is the difference between the degree of a term and the degree of a polynomial?

Tip: The degree of a polynomial is the highest sum of the exponents of variables in any single term.