Whether a function f(x) is increasing or decreasing at, say x=a, is indicated by the value of (df)/(dx) at x=a. If it is positive the function is increasing and if it is negative, the function is decreasing.
As f(x)=(-x^2+3x+2)/(x^2-1), using quotient rule
(df)/(dx)=((-2x+3)(x^2-1)-2x(-x^2+3x+2))/(x^2-1)^2
= (-2x^3+2x+3x^2-3+2x^3-6x^2-4x)/(x^2-1)^2
= (-3x^2-2x-3)/(x^2-1)^2
and at x=2, we have (df)/(dx)=(-3xx2^2-2xx2-3)/(2^2-1)^2=-19/9
Hence as (df)/(dx) is negative at x=2, f(x) is decreasing at x=2
graph{(-x^2+3x+2)/(x^2-1) [-8.62, 11.38, -4.16, 5.84]}