17. The equation of the directrix of the hyperbola $x(y+1)=1$ is
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17. $y=-x+\sqrt{2}-1$ or $y=-x-\sqrt{2}-1$.

The hyperbola $x(y+1)=1$ can be viewed as the hyperbola $xy=1$ translated one unit downward along the y-axis. The equations of the two directrices of $xy=1$ are $y=-x+\sqrt{2}$ or $y=-x-\sqrt{2}$. Translating these two lines downward by 1 unit gives the equations of the directrices of $x(y+1)=1$.