1. The solution set of the inequality $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$ is
$$
\text { - 1. }\left[-\frac{1}{2}, 0\right) \cup\left(0, \frac{45}{8}\right) \text {. }
$$

From $1-\sqrt{1+2 x} \neq 0$
$$
\Rightarrow x \geqslant-\frac{1}{2}, x \neq 0 \text {. }
$$

The original inequality becomes
$$
\begin{array}{l}
(1+\sqrt{1+2 x})^{2}<2 x+9 \\
\Rightarrow x<\frac{45}{8} .
\end{array}
$$

Therefore, the solution set of the original inequality is $\left[-\frac{1}{2}, 0\right) \cup\left(0, \frac{45}{8}\right)$.