Answer
Domain = $\displaystyle \mathbb{R}\backslash\{-1,\frac{3}{2}\}$
or $\{x\in \mathbb{R}\ |\ x\neq-1$ and $x\displaystyle \neq\frac{3}{2}\}$
Work Step by Step
The domain of a real function ( a function defined on real numbers)
is the set of numbers for which f(x) is defined.
$ f(x)=\displaystyle \frac{x+5}{2x^{2}-x-3}\quad$ can be calculated for any real number,
except for those yielding a zero in the denominator.
Here,
$ 2x^{2}-x-3=0\quad$ ... factor: two factors of $2(-3)=-6$ with sum $-1$ ...
$2x^{2}-3x+2x-3=0$
$x(2x-3)+(2x-3)=0$
$(2x-3)(x+1)=0$
$x=-1$ and $x=\displaystyle \frac{3}{2}\quad $must be excluded from the domain.
Domain = $\displaystyle \mathbb{R}\backslash\{-1,\frac{3}{2}\}$ or $\{x\in \mathbb{R}\ |\ x\neq-1$ and $x\displaystyle \neq\frac{3}{2}\}$