To multiply \( (7 - \sqrt{2})(-2 - \sqrt{2}) \), we can use the distributive property (also known as the FOIL method for binomials):
\[ (7 - \sqrt{2})(-2 - \sqrt{2}) = 7 \cdot (-2) + 7 \cdot (-\sqrt{2}) - \sqrt{2} \cdot (-2) - \sqrt{2} \cdot (-\sqrt{2}) \]
Calculating 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 the constant terms and the terms with \(\sqrt{2}\):
\[ (-14 + 2) + (-7\sqrt{2} + 2\sqrt{2}) = -12 - 5\sqrt{2} \]
So the exact answer is:
\[ -12 - 5\sqrt{2} \]
However, this specific answer is not among the options provided. It seems there may have been an oversight. Among the given options, none match the calculated result exactly. The response closest to a definitive computed answer would be selecting the one that has a negative square root term if you had to choose.
But based on the computed mathematical product of \( -12 - 5\sqrt{2} \), please ensure that you cross-check the question and provided responses. If needed, let me know if further calculations or clarifications are needed!