To multiply \((7 - \sqrt{2})(-2 - \sqrt{2})\), we can use the distributive property (also known as the FOIL method for binomials). Let's perform the multiplication step by step:
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First, multiply the first terms: \[ 7 \cdot (-2) = -14 \]
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Next, multiply the outer terms: \[ 7 \cdot (-\sqrt{2}) = -7\sqrt{2} \]
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Now, multiply the inner terms: \[ -\sqrt{2} \cdot (-2) = 2\sqrt{2} \]
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Finally, multiply the last terms: \[ -\sqrt{2} \cdot (-\sqrt{2}) = 2 \]
Now, combine all these results together: \[ -14 - 7\sqrt{2} + 2 + 2\sqrt{2} \]
Next, combine like terms:
- The constant terms: \(-14 + 2 = -12\)
- The terms with \(\sqrt{2}\): \(-7\sqrt{2} + 2\sqrt{2} = -5\sqrt{2}\)
Thus, the final result is: \[ -12 - 5\sqrt{2} \]
None of the options provided exactly matches this answer, but if we were to express it in a simplified format, it would be \(-12 - 5\sqrt{2}\). If you see an option that closely resembles this (particularly in terms of square roots combined with constants), that would be the correct response.
However, based on the given options:
- None of them are a perfect match to \(-12 - 5\sqrt{2}\).
If we were to round up the operation, the closest representation for this would not be accurately covered in the options provided.