To subtract the polynomial expressions \((3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7)\), we will distribute the negative sign across the second polynomial and then combine like terms.
First, distribute the negative sign:
\[ (3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7) = 3x^5 + 7x^2 - 4x + 8 - 4x^4 - 8x^2 + 3x - 7 \]
Now, combine like terms:
- \(x^5\) term: \(3x^5\)
- \(x^4\) term: \(-4x^4\)
- \(x^2\) terms: \(7x^2 - 8x^2 = -1x^2\)
- \(x\) terms: \(-4x + 3x = -1x\)
- Constant terms: \(8 - 7 = 1\)
Putting it all together, we have:
\[ 3x^5 - 4x^4 - 1x^2 - 1x + 1 \]
We can write it more neatly:
\[ 3x^5 - 4x^4 - x^2 - x + 1 \]
Thus, the simplified expression is:
\[ \boxed{3x^5 - 4x^4 - x^2 - x + 1} \]
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