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The problem provided is to subtract two polynomial expressions and simplify completely:

\[
(3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7)
\]

### Steps to Solve:

1. **Distribute the negative sign** to the second polynomial:
   \[
   3x^5 + 7x^2 - 4x + 8 - 4x^4 - 8x^2 + 3x - 7
   \]

2. **Group like terms**:
   \[
   3x^5 - 4x^4 + (7x^2 - 8x^2) + (-4x + 3x) + (8 - 7)
   \]

3. **Simplify each group**:
   - \(3x^5\) (no like term for \(x^5\))
   - \(-4x^4\) (no like term for \(x^4\))
   - \(7x^2 - 8x^2 = -x^2\)
   - \(-4x + 3x = -x\)
   - \(8 - 7 = 1\)

4. **Final simplified expression**:
   \[
   3x^5 - 4x^4 - x^2 - x + 1
   \]

This is the fully simplified expression.

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### Related questions:
1. How do you add and subtract polynomials with multiple variables?
2. What are the rules for distributing negative signs across polynomials?
3. How do you combine like terms when working with polynomials?
4. How does subtracting polynomials differ from adding them?
5. What is the general form of a polynomial, and how do you determine the degree?

**Tip**: Always remember to distribute negative signs carefully when subtracting polynomials.

Subtract the polynomial expressions and simplify completely: (3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7)

Learn how to subtract and simplify polynomial expressions with this example: (3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7). This step-by-step solution involves distributing the negative sign, grouping like terms, and simplifying the expression. Ideal for students in Grades 9-11.

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