To multiply the expressions \(4a^6 b^5 c^4\) and \(-7b^7 c^6\), you can follow these steps:
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Multiply the coefficients: \(4\) and \(-7\): \[ 4 \cdot -7 = -28 \]
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Multiply the variables with the same base by adding their exponents:
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For \(b\): \[ b^5 \cdot b^7 = b^{5+7} = b^{12} \]
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For \(c\): \[ c^4 \cdot c^6 = c^{4+6} = c^{10} \]
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The variable \(a\) only comes from the first term, so it remains \(a^6\).
Now, combine all parts together: \[ 4a^6 b^5 c^4 \cdot -7b^7 c^6 = -28a^6 b^{12} c^{10} \]
Thus, the result is: \[ \boxed{-28a^6 b^{12} c^{10}} \]