Final answer:
The vertices of the hyperbola are at (1, 2) and (-3, 2), and the foci are located at (-4, 2) and (2, 2).
Step-by-step explanation:
To find the vertices and foci of a hyperbola, let's examine the given equation (x+1)^2/4 - (y-2)^2/5 = 1.
First, we can identify that this is a hyperbola centered at (-1, 2) because the x-term is positive and the y-term is negative when set to equal 1. To find the vertices, we look at the coefficients of the squared terms. Since the x-term's coefficient is 4, the distance from the center to the vertices along the x-axis is 2 units (the square root of 4). Therefore, the vertices are at (-1 ± 2, 2), or (1, 2) and (-3, 2).
The distance between the foci and the center is denoted by 'c', which we can find using the formula c = √(a^2 + b^2), where 'a' is the distance to the vertices (2 in this case) and 'b' is the square root of the absolute value of the y-term's coefficient (so b = √5). Calculating c, we get c = √(4 + 5) = √9 = 3. Therefore, the foci are at (-1 ± 3, 2), or (-4, 2) and (2, 2).