To simplify the expression \((3i + 4)(-2i - 5) + i\), we will first distribute the terms in the parentheses.
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Distribute \((3i + 4)\) and \((-2i - 5)\):
\[ (3i + 4)(-2i) + (3i + 4)(-5) \]
- For \((3i + 4)(-2i)\):
\[ 3i \cdot (-2i) + 4 \cdot (-2i) = -6i^2 - 8i \]
- For \((3i + 4)(-5)\):
\[ 3i \cdot (-5) + 4 \cdot (-5) = -15i - 20 \]
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Now add these results together:
\[ -6i^2 - 8i - 15i - 20 = -6i^2 - 23i - 20 \]
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Recall that \(i^2 = -1\):
\[ -6(-1) - 23i - 20 = 6 - 23i - 20 \]
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Simplifying the constants:
\[ 6 - 20 = -14 \]
Putting it all together, we have:
\[ -14 - 23i \]
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Finally, add \(i\):
\[ -14 - 23i + i = -14 - 22i \]
Thus, the simplified expression is:
\[ \boxed{-14 - 22i} \]