10. Given the function $f(x)=x^{2}+x-2$.
(1) Try to find the analytical expression of the function $g(x)=\frac{|f(x)|-f(x)}{2}$;
(2) If $a>0$, the line $y=ax+b$ intersects the curve $y=g(x)$ at three different points, try to determine the relationship between $a$ and $b$, and draw a diagram to represent the region where the point $(a, b)$ lies with $a, b$ as coordinates.
$$
\left\{\begin{array}{l}
-20, \\
-20, \\
\varphi(1)>0 .
\end{array}\right.
$$

i.e., $\left\{\begin{array}{l}b<2 a, \\ b>-a .\end{array}\right.$
Given that $a>0$, therefore, $2 a<b<\frac{1}{4}(a+1)^{2}+2$, and $0<a<3$ is the solution.
The point $(a, b)$ with coordinates $a$ and $b$ lies in the shaded area of the graph on the right (excluding the boundary).