2. (HUN) For which real numbers $x$ does the following inequality hold:  $$ \frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9 ? $$
2. The LHS term is well-defined for $x \geq-1 / 2$ and $x \neq 0$. Furthermore, $4 x^{2} /(1-\sqrt{1+2 x})^{2}=(1+\sqrt{1+2 x})^{2}$. Since $f(x)=(1+\sqrt{1+2 x})^{2}-2 x-$ $9=2 \sqrt{1+2 x}-7$ is increasing and since $f(45 / 8)=0$, it follows that the inequality holds precisely for $-1 / 2 \leq x<45 / 8$ and $x \neq 0$.