To multiply \((7 - \sqrt{2})(-2 - \sqrt{2})\), we can use the distributive property (also known as the FOIL method for binomials):
\[ (7 - \sqrt{2})(-2 - \sqrt{2}) = 7 \cdot (-2) + 7 \cdot (-\sqrt{2}) - \sqrt{2} \cdot (-2) - \sqrt{2} \cdot (-\sqrt{2}) \]
Now calculate each term:
- \(7 \cdot (-2) = -14\)
- \(7 \cdot (-\sqrt{2}) = -7\sqrt{2}\)
- \(-\sqrt{2} \cdot (-2) = 2\sqrt{2}\)
- \(-\sqrt{2} \cdot (-\sqrt{2}) = 2\)
Now combine these results:
\[ -14 - 7\sqrt{2} + 2\sqrt{2} + 2 \]
Combine like terms:
The constant terms: \(-14 + 2 = -12\)
The terms with \(\sqrt{2}\): \(-7\sqrt{2} + 2\sqrt{2} = -5\sqrt{2}\)
Putting it all together, we have:
\[ -12 - 5\sqrt{2} \]
Thus, the exact answer is:
\(-12 - 5\sqrt{2}\) (not among the given options). If we are limited to the responses provided and need to match responses, the one that reflects a negative value would be the closest, but it does not accurately represent the computed answer. There might be an issue with the responses given, as none match.
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