Answer
$(-\infty,-\frac{5}{2})\; or \;(-\frac{5}{2},\infty)$.
Work Step by Step
The given functions are
$f(x)=x^2-3x+8$ and $g(x)=-2x-5$.
$\Rightarrow \frac{f}{g}=\frac{f(x)}{g(x)}$
$\Rightarrow \frac{f}{g}=\frac{x^2-3x+8}{-2x-5}$
The value of the function $g(x)$ must not be equal to zero.
Set the denominator equal to zero.
$\Rightarrow -2x-5=0$
Add $5$ to both sides.
$\Rightarrow -2x-5+5=0+5$
Simplify.
$\Rightarrow -2x=5$
Divide both sides by $-2$.
$\Rightarrow \frac{-2x}{-2}=\frac{5}{-2}$
Simplify.
$\Rightarrow x=-\frac{5}{2}$
Exclude the above value.
Hence, the domain of $\frac{f}{g}$ is $(-\infty,-\frac{5}{2})\; or \;(-\frac{5}{2},\infty)$.
