Answer
All real numbers except $-\frac{1}{2}$ and $3$.
Work Step by Step
Factor the denominator to obtain:
$2x^2-5x-3=(2x+1)(x-3)$
The denominator cannot be $0$ hence,
$(2x+1)(x-3)\ne0\\
\longrightarrow 2x+1\ne0, \text{ and } x-3\ne0\\
\longrightarrow x\ne-\frac{1}{2}\text{ and } x\ne3$.
Therefore the domain is all real numbers except $-\frac{1}{2}$ and $3$.