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 $y^2/9+2y/3+1-x^2/16+x/2-1=1$.