1. **Problem statement:** Calculate the value of $\left(\frac{1}{\sqrt{2}} - 1\right)^{-1}$. 2. **Formula and rules:** The expression is a reciprocal (inverse) of $\frac{1}{\sqrt{2}} - 1$. To simplify, first simplify the denominator, then take the reciprocal. 3. **Simplify the denominator:** $$\frac{1}{\sqrt{2}} - 1 = \frac{1}{\sqrt{2}} - \frac{\sqrt{2}}{\sqrt{2}} = \frac{1 - \sqrt{2}}{\sqrt{2}}$$ 4. **Take the reciprocal:** $$\left(\frac{1 - \sqrt{2}}{\sqrt{2}}\right)^{-1} = \frac{\sqrt{2}}{1 - \sqrt{2}}$$ 5. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate $1 + \sqrt{2}$: $$\frac{\sqrt{2}}{1 - \sqrt{2}} \times \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{\sqrt{2}(1 + \sqrt{2})}{(1)^2 - (\sqrt{2})^2} = \frac{\sqrt{2} + 2}{1 - 2} = \frac{\sqrt{2} + 2}{-1} = - (\sqrt{2} + 2)$$ 6. **Final answer:** $$-(2 + \sqrt{2})$$ This corresponds to option (D).