Let's solve the problems step by step:

1. Add the given expressions:

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

Step 1: Combine like terms. Rearrange the terms by degree for clarity:

• Terms with $x^4$: $-3x^4 + 7x^4 = 4x^4$
• Terms with $x^3$: $8x^3 - 9x^3 = -x^3$
• Terms with $x^2$: $4x^2 + 5x^2 = 9x^2$
• Terms with $x$: $-x$
• Constant terms: $7$

Result: $4x^4 - x^3 + 9x^2 - x + 7$

2. Subtract the given expressions:

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

Step 1: Distribute the subtraction across the second set of parentheses: $9x + 3x^2 - 1 - 4 + 2x - 5x^2 - 6x^3$

Step 2: Combine like terms:

• Terms with $x^3$: $-6x^3$
• Terms with $x^2$: $3x^2 - 5x^2 = -2x^2$
• Terms with $x$: $9x + 2x = 11x$
• Constant terms: $-1 - 4 = -5$

Result: $-6x^3 - 2x^2 + 11x - 5$

Final Answers:

1. $4x^4 - x^3 + 9x^2 - x + 7$
2. $-6x^3 - 2x^2 + 11x - 5$

Would you like a more detailed explanation for any step, or do you have further questions?

Expanding Learning:

1. Can you explain why combining like terms is valid in polynomial addition/subtraction?
2. How would the process change if multiplication were involved instead of addition or subtraction?
3. What happens to terms with different variables (e.g., $x$ and $y$) during addition or subtraction?
4. How do you determine the degree of a polynomial?
5. Why is it important to distribute the subtraction sign correctly?

Tip: Always align like terms vertically when adding or subtracting polynomials to avoid mistakes.