To determine whether the function f(x) = (x^2-2x-3)/(x^2+3x+2) has any holes, we can factor the numerator and denominator and simplify the expression. The numerator can be factored as:
x^2 - 2x - 3 = (x - 3)(x + 1)
And the denominator can be factored as:
x^2 + 3x + 2 = (x + 1)(x + 2)
Therefore, we can simplify the function as:
f(x) = [(x - 3)(x + 1)]/[(x + 1)(x + 2)]
The factor of (x + 1) appears in both the numerator and denominator, so we can simplify further by canceling it out:
f(x) = (x - 3)/(x + 2)
Since (x + 1) was canceled out, we have a hole in the graph of the original function at x = -1. To find the coordinates of the hole, we can evaluate the simplified function at x = -1:
f(-1) = (-1 - 3)/(-1 + 2) = -4
Therefore, the hole in the graph of the original function is located at the point (-1, -4).