To find the sum or difference of the given expression, we will combine the terms step by step:

Expression:

$(9x^4 - 4x^2 + x) - (8x^3 + 10x^2 + 5x) + (x^2 - x)$

1. Distribute the negative signs for each term inside the parentheses: $= 9x^4 - 4x^2 + x - 8x^3 - 10x^2 - 5x + x^2 - x$

2. Combine like terms by grouping terms with the same powers of $x$:

   • $x^4$ term: $9x^4$
   • $x^3$ term: $-8x^3$
   • $x^2$ terms: $-4x^2 - 10x^2 + x^2 = -13x^2$
   • $x$ terms: $x - 5x - x = -5x$

3. Rewrite the simplified expression: $= 9x^4 - 8x^3 - 13x^2 - 5x$

Final Answer:

$9x^4 - 8x^3 - 13x^2 - 5x$

Would you like additional explanation on each step or have further questions?

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Related Questions to Explore:

1. How do you distribute negative signs in algebraic expressions?
2. What are like terms, and why are they important in simplification?
3. How does combining terms help in solving polynomial equations?
4. What is the degree of the polynomial $9x^4 - 8x^3 - 13x^2 - 5x$?
5. How would adding or subtracting constants affect the polynomial simplification?

Tip: Always align terms with similar powers to simplify polynomials accurately.