`\color{red}{@\text{Mon}}`

`\text{ Rút gọn:}`

`P=\frac{\sqrt{x}+1}{\sqrt{x}+2}+\frac{2\sqrt{x}}{\sqrt{x}+2}+\frac{5\sqrt{x}+2}{4-x}`

`=\frac{(\sqrt{x}+1)(\sqrt{x}+2)+2\sqrt{x}(\sqrt{x}-2)-5\sqrt{x}-2}{(\sqrt{x}-2)(\sqrt{x}+2)}`

`=\frac{x+3\sqrt{x}+2+2x-4\sqrt{x}-5\sqrt{x}-2}{(\sqrt{x}-2)(\sqrt{x}+2)}`

`=\frac{3x-6\sqrt{x}}{(\sqrt{x}-2)(\sqrt{x}+2)}`

`=\frac{3\sqrt{x}(\sqrt{x}-2)}{(\sqrt{x}-2)(\sqrt{x}+2)}`

`=\frac{3\sqrt{x}}{\sqrt{x}+2}`

`4)P=\frac{3\sqrt{x}}{\sqrt{x}+2}`

`=\frac{3(\sqrt{x}+2)-6}{\sqrt{x}+2}`

`=3-\frac{6}{\sqrt{x}+2}`

`Để` `P in ZZ:`

`<=>\sqrt{x}+2 in Ư(6)={-6;-3;-2;-1;1;2;3;6}`

`<=>\sqrt{x} in {0;1;4}(Do` `\sqrt{x} >=0)`

`<=>x in {0;1;16}`

`5)P=3-\frac{6}{\sqrt{x}+2}`

`Do` `\sqrt{x} >= 0`

`=>\sqrt{x}+2 >= 2`

`=>\frac{6}{\sqrt{x}+2} <= \frac{6}{2}=3`

`=>3-\frac{6}{\sqrt{x}+2}>=3-3=0` `hay` `P>=0`

`\text{ Dấu "=" xảy ra khi}`

`<=>\sqrt{x}=0`

`<=>x=0`

`\text{ Vậy GTNN của P = 0 }`

`6)x-17\sqrt{x}+90-2P.(\sqrt{x}+2)=0`

`<=>x-17\sqrt{x}+90-2..\frac{3\sqrt{x}}{\sqrt{x}+2}.(\sqrt{x}+2)=0`

`<=>x-17\sqrt{x}+90-6\sqrt{x}=0`

`<=>x-23\sqrt{x}+90=0`

`<=>(\sqrt{x}-18)(\sqrt{x}-5)=0`

`<=>` \(\left[ \begin{array}{l}\sqrt{x}=18\\\sqrt{x}=5\end{array} \right.\) 

`<=>`\(\left[ \begin{array}{l}x=324(tm)\\x=25(tm)\end{array} \right.\) 

$\displaystyle \begin{array}{{>{\displaystyle}l}} P=\frac{\sqrt{x} +1}{\sqrt{x} -2} +\frac{2\sqrt{x}}{\sqrt{x} +2} +\frac{5\sqrt{x} +2}{4-x} \ \\ x\#4\\ P=\frac{\left(\sqrt{x} +1\right)\left(\sqrt{x} +2\right) +2\sqrt{x}\left(\sqrt{x} -2\right) -5\sqrt{x} -2}{\left(\sqrt{x} -2\right)\left(\sqrt{x} +2\right) \ }\\ P=\frac{x+3\sqrt{x} +2+2x-4\sqrt{x} -5\sqrt{x} -2}{\left(\sqrt{x} -2\right)\left(\sqrt{x} +2\right)}\\ P=\frac{3x-6\sqrt{x}}{\left(\sqrt{x} -2\right)\left(\sqrt{x} +2\right)} =\frac{2\sqrt{x}\left(\sqrt{x} -2\right)}{\left(\sqrt{x} -2\right)\left(\sqrt{x} +2\right)} =\frac{3\sqrt{x}}{\sqrt{x} +2} \ \\ 2) \ x=6-4\sqrt{2} =2-2.2\sqrt{2} +4=\left( 2-\sqrt{2}\right)^{2} \ \\ \rightarrow \ \sqrt{x} =2-\sqrt{2} \ \\ P=\frac{3\left( 2-\sqrt{2}\right)}{2-\sqrt{2} +2} =\frac{6-3\sqrt{2}}{4-\sqrt{2}}\\ 3) \ P >2\ \rightarrow \ P-2 >0\ \\ Do\ đó\ :\ \frac{3\sqrt{x}}{\sqrt{x} +2} \ -2 >0\ \\ \rightarrow \ \frac{3\sqrt{x} -2\sqrt{x} -4}{\sqrt{x} +2} \ >0\ \\ \rightarrow \frac{\sqrt{x} -4}{\sqrt{x} +2} \ >0\ \\ \rightarrow \begin{cases} \left[ \begin{array}{l l} \sqrt{x} -4 >0 & \\ \sqrt{x} +2 >0 & \end{array} \right. & \\ \left[ \begin{array}{l l} \sqrt{x} -4< 0 & \\ \sqrt{x} +2< 0\ & \end{array} \right. & \end{cases}\rightarrow \begin{cases} \left[ \begin{array}{l l} x >16 & \\ x\ thõa\ mãn\ với\ mọi\ dk & \end{array} \right. & \\ \left[ \begin{array}{l l} x< 16 & \\ vô\ nghiệm\ & \end{array} \right. & \end{cases}\\ \rightarrow \ x< 16;x< 0\ \\ 4) \ P=\frac{3\sqrt{x}}{\sqrt{x} +2} =\ \ 3-\frac{6}{\sqrt{x} +2} \ \\ Vì\ x\ nguyên\ nên\ P\ nguyên\ khi\ \sqrt{x} +2\ là\ ước\ của\ 6\ \\ và\ vì\ \sqrt{x} +2\geqslant 2\ nên\ \sqrt{x} +2\ thuộc\ \{6;3;2\}\\ Với\ \sqrt{x} +2=3\ \rightarrow x=1\ \\ \sqrt{x} +2=6\ \rightarrow x=16\ \\ \sqrt{x} +2=2\ \rightarrow x=0\ \\ Vậy.......\ \\ 5) \ P=\frac{3\sqrt{x}}{\sqrt{x} +2} \ \\ Nhận\ thấy\ 3\sqrt{x} \geqslant 0\ \ và\ \sqrt{x} +2 >2\\ Do\ đó\ :\ P=\frac{2\sqrt{x}}{\sqrt{x} +2} \geqslant 0\ .\ Dấu\ =\ xảy\ ra\ khi\ x=0\ \\ Vậy\ minP=0\ Tại\ x=0\ \\ 6) \ x-17\sqrt{x} +90-2\frac{3\sqrt{x}}{\sqrt{x} +2}\left(\sqrt{x} +2\right) =0\ \\ DK:\ x\ \geqslant 0\ \\ x-17\sqrt{x} +90-6\sqrt{x} =0\ \\ x-23\sqrt{x} +90=0\ \\ x-5\sqrt{x} -18\sqrt{x} +90=0\ \\ \rightarrow \sqrt{x}\left(\sqrt{x} -5\right) -18\left(\sqrt{x} -5\right) =0\ \\ \left(\sqrt{x} -5\right)\left(\sqrt{x} -18\right) =0\ \\ \rightarrow \left[ \begin{array}{l l} \sqrt{x} -5=0 & \\ \sqrt{x} -18=0 & \end{array} \right.\rightarrow \left[ \begin{array}{l l} \sqrt{x} =5 & \\ \sqrt{x} =18\ & \end{array} \right.\rightarrow \left[ \begin{array}{l l} x=25\ ( tm) & \\ x=324\ ( \ tm) \ & \end{array} \right.\\ Vậy\ x=25\ hoặc\ x=324\ \\ \\ \end{array}$