To re-express \(2 \sqrt{-45}\) as a complex number in the form \(a + bi\), we first simplify the expression.
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Start with the square root of \(-45\):
\[ \sqrt{-45} = \sqrt{45} \cdot \sqrt{-1} \]
Here, \(\sqrt{-1}\) is represented as \(i\) (the imaginary unit).
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Next, calculate \(\sqrt{45}\):
\[ \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \]
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Now substitute back into the expression:
\[ \sqrt{-45} = 3\sqrt{5} \cdot i \]
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Multiply by 2 to get \(2 \sqrt{-45}\):
\[ 2 \sqrt{-45} = 2 \cdot 3\sqrt{5} \cdot i = 6\sqrt{5} \cdot i \]
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Finally, express it in the form \(a + bi\):
\[ 2 \sqrt{-45} = 0 + 6\sqrt{5} i \]
Thus, the complex number in the form \(a + bi\) is:
\[ \boxed{0 + 6\sqrt{5} i} \]