To simplify the expression \( (3i - 5)[(i + 2) + (2i - 4)] \), we will first simplify the expression inside the brackets.
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Combine the terms inside the brackets: \[ (i + 2) + (2i - 4) = i + 2 + 2i - 4 = (i + 2i) + (2 - 4) = 3i - 2 \]
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Now substitute this back into the original expression: \[ (3i - 5)(3i - 2) \]
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Next, we will distribute \( (3i - 5) \) over \( (3i - 2) \): \[ = 3i(3i) + 3i(-2) - 5(3i) - 5(-2) \] \[ = 9i^2 - 6i - 15i + 10 \]
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Combine like terms. Remember that \( i^2 = -1 \): \[ = 9(-1) - 6i - 15i + 10 \] \[ = -9 - 21i + 10 \] \[ = 1 - 21i \]
Thus, the simplified expression is: \[ \boxed{1 - 21i} \]