To subtract the polynomial expressions \( (3x^5 + 7x^2 - 4x + 8) - (4x^4 + 8x^2 - 3x + 7) \), we first rewrite the expression by distributing the negative sign to the second polynomial:
\[ 3x^5 + 7x^2 - 4x + 8 - 4x^4 - 8x^2 + 3x - 7 \]
Next, we combine like terms. We will arrange the terms by their degree:
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The \( x^5 \) term:
- \( 3x^5 \)
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The \( x^4 \) term:
- \( -4x^4 \)
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The \( x^2 \) terms:
- \( 7x^2 - 8x^2 = -x^2 \)
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The \( x \) terms:
- \( -4x + 3x = -x \)
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The constant terms:
- \( 8 - 7 = 1 \)
Now, we can combine all of the simplified terms:
\[ 3x^5 - 4x^4 - x^2 - x + 1 \]
Thus, the simplified expression is:
\[ \boxed{3x^5 - 4x^4 - x^2 - x + 1} \]
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