Introduction:
To find the equation of a hyperbola, we need to know the coordinates of its foci and vertices. In this case, the foci are given as (0, -3) and the vertices as (0, -√(11/2)). Let's proceed to find the equation of the hyperbola.

Finding the center:
The center of the hyperbola is the midpoint between the foci. So, the center of the hyperbola is (0, (-3 + (-√(11/2)))/2) = (0, (-3 - √(11/2))/2).

Finding the distance between the center and the foci:
The distance between the center and the foci is given by the formula c = √(a^2 + b^2), where a is the distance between the center and vertex, and b is the distance between the center and the co-vertex. In this case, the distance between the center and vertex is √(11/2), and the distance between the center and the co-vertex is 3. Therefore, c = √((√(11/2))^2 + 3^2) = √(11/2 + 9) = √(11/2 + 18/2) = √(29/2).

Finding the value of a:
The value of a is the distance between the center and vertex, which is √(11/2).

Finding the value of b:
The value of b is the distance between the center and the co-vertex, which is 3.

Equation of a hyperbola:
The equation of a hyperbola with center (h, k), semi-major axis a, and semi-minor axis b is given by:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

In this case, the center is (0, (-3 - √(11/2))/2), a = √(11/2), and b = 3. Substituting these values into the equation, we get:

(x - 0)^2 / (√(11/2))^2 - (y - ((-3 - √(11/2))/2))^2 / 3^2 = 1

Simplifying the equation further:

x^2 / (11/2) - (y + (3 + √(11/2))/2)^2 / 9 = 1

Multiplying both sides of the equation by 2, we get:

2x^2 / 11 - 2(y + (3 + √(11/2))/2)^2 / 9 = 2

Therefore, the equation of the hyperbola is:

2x^2 / 11 - 2(y + (3 + √(11/2))/2)^2 / 9 = 2