Simplifying the given expressions:

1. Add

$(8x^3 + 4x^2 - 3x^4 + 7) + (5x^2 + 7x^4 - 9x^3 - x)$

Group like terms by their powers of $x$: $= (-3x^4 + 7x^4) + (8x^3 - 9x^3) + (4x^2 + 5x^2) + (7 - x)$

Simplify each group: $= 4x^4 - x^3 + 9x^2 + 7 - x$

Final Answer: $4x^4 - x^3 + 9x^2 - x + 7$

2. Subtract

$(9x + 3x^2 - 1) - (4 - 2x + 5x^2 + 6x^3)$

Distribute the negative sign: $= (9x + 3x^2 - 1) - 4 + 2x - 5x^2 - 6x^3$

Group like terms by their powers of $x$: $= -6x^3 + (3x^2 - 5x^2) + (9x + 2x) + (-1 - 4)$

Simplify each group: $= -6x^3 - 2x^2 + 11x - 5$

Final Answer: $-6x^3 - 2x^2 + 11x - 5$

Would you like further clarification or a step-by-step breakdown?

Follow-Up Questions:

1. Can you expand polynomials with multiple variables?
2. How do you determine the degree of a polynomial after simplification?
3. What are common mistakes when combining like terms?
4. How is subtracting polynomials different from adding them?
5. Can you explain how to factor a polynomial once simplified?

Tip:

When combining like terms, carefully match the coefficients of the same powers of $x$ to avoid mistakes.