To re-express the negative square root \(-\sqrt{40}\) as a complex number in standard form \(a + bi\), we start by recognizing that the square root of a negative number can be expressed with the imaginary unit \(i\), where \(i = \sqrt{-1}\).
First, we simplify \(-\sqrt{40}\):
\[ -\sqrt{40} = -\sqrt{4 \cdot 10} = -\sqrt{4} \cdot \sqrt{10} = -2\sqrt{10} \]
Since \(2\sqrt{10}\) is a real number and we need to express it as an imaginary number, we rewrite it as:
\[ -2\sqrt{10} = 0 - 2\sqrt{10}i \]
Now, it's in the form \(a + bi\) where \(a = 0\) and \(b = -2\sqrt{10}\).
Now, let's match this to the responses provided:
- \(2i\sqrt{10}\) - This is not correct, as it is positive.
- \(4i\sqrt{10}\) - This is also positive.
- \(2\sqrt{10}\) - This is a positive real number.
- \(2\sqrt{10}i\) - This is positive as well.
None of these options appears to represent \(-2\sqrt{10}\) correctly, but it seems likely the closest should have a negative imaginary part.
Thus, none of the given options correctly represent \(-\sqrt{40}\) as a complex number.
If you need a precise selection based on the options available and given that these options seem inconsistent, please verify if there are alternative choices or if you require further assistance.
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