To subtract the expression \((3b - 2c) - (6c + 2b) - (b + c)\), we can start by distributing the negative signs and combining like terms.
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Distribute the negative signs: \[ (3b - 2c) - (6c + 2b) - (b + c) = 3b - 2c - 6c - 2b - b - c \]
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Now combine the like terms. First, let's combine the \(b\) terms: \[ 3b - 2b - b = (3 - 2 - 1)b = 0b = 0 \]
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Next, combine the \(c\) terms: \[ -2c - 6c - c = (-2 - 6 - 1)c = -9c \]
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Putting it all together: \[ 0 - 9c = -9c \]
Thus, the result of the expression \((3b - 2c) - (6c + 2b) - (b + c)\) is \(-9c\).