Factor the numerator and simplify the ratio to determine the location of holes and/or vertical asymptotes.
No holes (the ratio does not simplify). Vertical asymptote at $x = 2$.

Evaluate values close to $x = 2$ (from the left and from the right) to determine if the function approaches its asymptote in an increasing or decreasing manner.
$y(2^{-})\approx y(1.9)=\frac{(1.9+5)(1.9-3)}{(1.9-2)}=\frac{(5.9)(-1.1)}{(-0.1)}=64.9$ going up from the left.
This will help you with the approach statements for $x → 2^+$ and $x → 2^−$.

Use long division to determine the end behavior.
This will also help you write the approach statements for $x → − ∞$ and $x → ∞$.

Use the graph to write your approach statements and your slope statement.