Answer
The oblique asymptote is $y=2x$.
Work Step by Step
To find the oblique asymptotes, we look to change the function form into the following: $$y=A+\frac{B}{C}$$
in which $A$ is our linear line, which is the oblique asymptote as well, and $B/C$ is the remainder.
$$y=\frac{2x^{3/2}+2x-3}{\sqrt x+1}$$
$$y=\frac{2x^{3/2}+2x}{\sqrt x+1}-\frac{3}{\sqrt x+1}$$
$$y=\frac{2x(x^{1/2}+1)}{\sqrt x+1}-\frac{3}{\sqrt x+1}$$
$$y=2x-\frac{3}{\sqrt x+1}$$
As $x\to\infty$, $\sqrt x+1$ approaches $\infty$ as well, so $-3/(\sqrt x+1)$ will approach $0$.
The function, as a result, will behave like the line $y=2x$. Therefore, $y=2x$ is the oblique asymptote of the function.
