To simplify the expression \((3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7)\), we need to distribute the negative sign across the second polynomial and then combine like terms.
Step 1: Distribute the negative sign
\[ (3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7) = 3x^5 + 7x^2 - 4x + 8 - 4x^4 - 8x^2 + 3x - 7 \]
Step 2: Combine like terms
Now, let's gather the like terms:
- \(x^5\) terms: \(3x^5\)
- \(x^4\) terms: \(-4x^4\)
- \(x^2\) terms: \(7x^2 - 8x^2 = -x^2\)
- \(x\) terms: \(-4x + 3x = -x\)
- Constant terms: \(8 - 7 = 1\)
Step 3: Write the final expression
Now putting together all the terms we gathered:
\[ 3x^5 - 4x^4 - x^2 - x + 1 \]
Therefore, the simplified expression is:
\[ \boxed{3x^5 - 4x^4 - x^2 - x + 1} \]