To subtract the polynomial expressions \((3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7)\), we need to distribute the negative sign to each term in the second polynomial and then combine like terms.
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Rewrite the expression: \[ 3x^5 + 7x^2 - 4x + 8 - (4x^4 + 8x^2 - 3x + 7) \]
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Distribute the negative sign: \[ 3x^5 + 7x^2 - 4x + 8 - 4x^4 - 8x^2 + 3x - 7 \]
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Combine like terms:
- For \(x^5\): \(3x^5\)
- For \(x^4\): \(-4x^4\)
- For \(x^2\): \(7x^2 - 8x^2 = -1x^2\)
- For \(x\): \(-4x + 3x = -1x\)
- For the constant terms: \(8 - 7 = 1\)
Putting it all together, we have: \[ 3x^5 - 4x^4 - x^2 - x + 1 \]
Thus, the simplified expression is: \[ \boxed{3x^5 - 4x^4 - x^2 - x + 1} \]