To simplify the expression \((−3i−5)[(i+2)+(2i−4)]\), we first simplify the expression inside the brackets.
- Combine the terms inside the brackets: \[ (i + 2) + (2i - 4) = i + 2 + 2i - 4 = (i + 2i) + (2 - 4) = 3i - 2. \]
Now, substitute this back into the original expression: \[ (-3i - 5)(3i - 2). \]
- Next, we distribute \((-3i - 5)\) with \((3i - 2)\): \[ (-3i)(3i) + (-3i)(-2) + (-5)(3i) + (-5)(-2). \]
Calculating each term:
- \((-3i)(3i) = -9i^2\)
- \((-3i)(-2) = 6i\)
- \((-5)(3i) = -15i\)
- \((-5)(-2) = 10\)
Now combine these results: \[ -9i^2 + 6i - 15i + 10. \]
This simplifies to: \[ -9i^2 - 9i + 10. \]
-
Since \(i^2 = -1\), replace \(-9i^2\) with \(9\): \[ 9 - 9i + 10. \]
-
Finally, combine like terms: \[ 9 + 10 - 9i = 19 - 9i. \]
Thus, the simplified expression is: \[ \boxed{19 - 9i}. \]