Answer
$$ y=2 x .$$
Work Step by Step
Since the highest power of $ x $ in the numerator is 1 more than the highest power of $ x $ in the denominator, there is an oblique asymptote.
$$ y=\frac{2 x^{3 / 2}+2 x-3}{\sqrt{x}+1}=2 x-\frac{3}{\sqrt{x}+1},$$
Thus the oblique asymptote is $$ y=2 x $$
