Find the expanded form of the hyperbola with the following properties: center at (4, -3), vertices (4, 1) and (4, -7), and foci at (4, 2) and (4, -8).

Step 1 ：Step 1: Calculate the values of a and c. The distance from the center to a vertex is a, and the distance from the center to a focus is c. Using the distance formula, we find that \(a = \sqrt{(4-4)^2 + (1--3)^2} = 4\) and \(c = \sqrt{(4-4)^2 + (2--3)^2} = 5\).

Step 2 ：Step 2: Use the relationship \(c^2 = a^2 + b^2\) to solve for b. Substituting the values we found for a and c, we get \(b = \sqrt{c^2 - a^2} = \sqrt{5^2 - 4^2} = 3\).

Step 3 ：Step 3: Write the standard form of the equation for a vertical hyperbola centered at (h, k) with a horizontal axis of length 2a and a vertical axis of length 2b: \((y-k)^2/b^2 - (x-h)^2/a^2 = 1\). Substituting the values we found for h (4), k (-3), a (4), and b (3), we get \((y+3)^2/3^2 - (x-4)^2/4^2 = 1\).

Step 4 ：Step 4: Expand the equation to get its expanded form. The expanded form is \(y^2/9 + 2y/3 + 1 - x^2/16 + x/2 - 1 = 1\).