To subtract the polynomials \((-6f^3 - 2f^2 - 8f + 1)\) and \((3f^3 + 6f^2 + 4)\), you need to change the sign of the second polynomial and then combine like terms.
The expression becomes:
\[ -6f^3 - 2f^2 - 8f + 1 - (3f^3 + 6f^2 + 4) \]
Distributing the negative sign gives:
\[ -6f^3 - 2f^2 - 8f + 1 - 3f^3 - 6f^2 - 4 \]
Now combine like terms:
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For \(f^3\): \(-6f^3 - 3f^3 = -9f^3\)
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For \(f^2\): \(-2f^2 - 6f^2 = -8f^2\)
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For \(f\): \(-8f\) has no like terms.
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For the constant: \(1 - 4 = -3\)
Putting it all together, the result is:
\[ -9f^3 - 8f^2 - 8f - 3 \]
Thus, the final result is:
\[ \boxed{-9f^3 - 8f^2 - 8f - 3} \]