10. Let $x, y \in \mathbf{R}$. If $2 x, 1, y-1$ form an arithmetic sequence, and $y+3,|x+1+| x-1 \mid$, arsin $\left.\sqrt{1-x^{2}}\right)$ form a geometric sequence, then the value of $(x+1)(y+1)$ is
10.4 or $2(\sqrt{17}-3)$

Given $-1 \leqslant x \leqslant 1$, and $|x| \cdot(y+3)=4, 2x+y=3$.
When $x \geqslant 0$, solving yields $x=1, y=1$, at this point $(x+1)(y+1)=4$.
When $x<0$, solving yields $x=\frac{3-\sqrt{17}}{2}, y=\sqrt{17}$, at this point $(x+1)(y+1)=2(\sqrt{17}-3)$.