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$M=(\frac{\sqrt{x}}{\sqrt{x}+6}+\frac{1}{\sqrt{x}-6}+\frac{17\sqrt{x}+30}{x-36}):\frac{\sqrt{x}+6}{24}$

$=(\frac{\sqrt{x}}{\sqrt{x}+6}+\frac{1}{\sqrt{x}-6}+\frac{17\sqrt{x}+30}{(\sqrt{x}-6).(\sqrt{x}+6)}):\frac{\sqrt{x}+6}{24}$

$=(\frac{\sqrt{x}(\sqrt{x}-6)+\sqrt{x}+6+17\sqrt{x}+30}{(\sqrt{x}-6)(\sqrt{x}+6)}):\frac{\sqrt{x}+6}{24}$

$= (\frac{x+12\sqrt{x}+36}{(\sqrt{x}-6)(\sqrt{x}+6)}):\frac{\sqrt{x}+6}{24}$

$= \frac{(\sqrt{x}+6)^{2}}{(\sqrt{x}-6)(\sqrt{x}+6)}:\frac{\sqrt{x}+6}{24}$

$= \frac{\sqrt{x}+6}{\sqrt{x}-6}:\frac{\sqrt{x}+6}{24}$

$= \frac{\sqrt{x}+6}{\sqrt{x}-6}.\frac{24}{\sqrt{x}+6}$

$= \frac{24}{\sqrt{x}-6}$

M lớn  nhất $

⇔$ \frac{24}{\sqrt{x}-6}$ ≤ $\frac{24}{-6}=-4$  

Dấu '=' xảy ra khi x=0

vậy  M = -4 khi x=0