The domain of definition of function $\mathrm{f}\left(\mathrm{x}\right)=\log \left(\sqrt{{\mathrm{x}}^2-5\mathrm{x}-24 } -\mathrm{x}-2\right),$is 

Solution:

Here,
$\sqrt{{\mathrm{x}}^2-5\mathrm{x}-24}>\mathrm{x}+2$ is equivalent to the collection of two system of inequations.
Case I
 $$\begin{gathered} {\mathrm{x}}^2-5\mathrm{x}-24\ge 0 \mathrm{and} \mathrm{x}+2<0 \\ \Rightarrow (\mathrm{x}-8)(\mathrm{x}+3)\ge 0 \mathrm{and} \mathrm{x}<-2 \\ \Rightarrow \mathrm{x}\le -3 .....\left(\mathrm{i}\right) \end{gathered}$$
Case II
$$\begin{gathered} {\mathrm{x}}^2-5\mathrm{x}-24\ge 0,\mathrm{x}+2\ge 0 \\ \mathrm{and} {\mathrm{x}}^2-5\mathrm{x}-24>{\left(\mathrm{x}+2\right)}^2 \\ \Rightarrow \mathrm{x}\in (-\infty ,-3]\cup [8,\infty ),\mathrm{x}\in [-2,\infty ) \\ \mathrm{and} {\mathrm{x}}^2-5\mathrm{x}-24>{\mathrm{x}}^2+4\mathrm{x}+4 \\ \Rightarrow 9\mathrm{x}<-28 \\ \Rightarrow \mathrm{x}<\frac{-28}{9} \\ \therefore \mathrm{No} \mathrm{solution} \mathrm{i}.\mathrm{e}., \mathrm{x}\in \mathrm{\varphi } ....\left(\mathrm{ii}\right) \\ \mathrm{From} \mathrm{Eqs}.\left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right), \mathrm{we} \mathrm{get} \mathrm{x}\in (-\infty ,-3] \end{gathered}$$