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To solve this, let's analyze the polynomial expression you provided:

\[
5x^4 + 4x^5 - 6y^5 + 2x^6
\]

### Steps to determine the degree of the polynomial:
1. The degree of a polynomial is the highest sum of the exponents of the variables in any single term of the polynomial.
   
2. Let's break down each term in the expression:
   - \( 5x^4 \) has a degree of 4 (since the exponent of \( x \) is 4).
   - \( 4x^5 \) has a degree of 5 (since the exponent of \( x \) is 5).
   - \( -6y^5 \) has a degree of 5 (since the exponent of \( y \) is 5).
   - \( 2x^6 \) has a degree of 6 (since the exponent of \( x \) is 6).

3. Among the terms, the highest degree is 6 from the term \( 2x^6 \).

### Conclusion:
The degree of the polynomial is **6**.

Would you like more details on this? Here are some related questions to expand on:

1. What is a polynomial, and how are degrees determined for more complex expressions?
2. How do coefficients affect the degree of a polynomial?
3. Can polynomials involve more than two variables, and how would that affect the degree calculation?
4. How do you find the degree of a polynomial when it contains both \( x \) and \( y \) terms?
5. What is the significance of the degree of a polynomial in real-world applications?

**Tip**: The degree of a polynomial tells us the behavior of its graph at extreme values of \( x \) and helps in predicting how the polynomial functions behave.

Enter the degree of the polynomial: 5x^4 + 4x^5 - 6y^5 + 2x^6

Learn how to determine the degree of the polynomial 5x^4 + 4x^5 - 6y^5 + 2x^6 by identifying the highest exponent sum of any term. This problem is essential for students studying Algebra and is suitable for Grades 8-10.

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