The x-intercepts lead to the conclusion that the numerator of the rational function must have the factors (x+3) and (x-1).
The fact that there is a vertical asymptote at x=-2 means there must be at least one factor of (x + 2) in the denominator.
In order to have a horizontal asymptote at y = 1 the degree of the numerator must equal the degree of the denominator, so we need to introduce a factor of x into the denominator, but without introducing another vertical asymptote. That leads to the conclusion that another factor of (x + 2) must appear in the denominator.
So far we have f(x) = (x2 + 2x - 3) / (x2 + 4x + 4)
As x approached +/- infinity f(x) approaches 1, so there is the horizontal asymptote. What is f(0)? Looks like -3/4, so all conditions are satisfied. See https://www.desmos.com/calculator/fuurmwqm4g for the graph.