\(P=\frac{3x+3\sqrt{x}-9}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\frac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{3x+3\sqrt{x}-9+x+2\sqrt{x}-3-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\frac{3x+5\sqrt{x}-8}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{\left(\sqrt{x}-1\right)\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\frac{3\sqrt{x}+8}{\sqrt{x}+2}\)
\(P< \frac{15}{4}\Rightarrow\frac{3\sqrt{x}+8}{\sqrt{x}+2}< \frac{15}{4}\)
\(\Rightarrow12\sqrt{x}+32< 15\sqrt{x}+30\Rightarrow3\sqrt{x}>2\)
\(\Rightarrow\sqrt{x}>\frac{2}{3}\Rightarrow x>\frac{4}{9}\Rightarrow\left\{{}\begin{matrix}x>\frac{4}{9}\\x\ne1\end{matrix}\right.\)
