Final answer:
The standard form of the equation for a hyperbola with specified vertices and foci is (y + 1)²/10 - (x + 2)²/c² = 1.
Step-by-step explanation:
The standard form of the equation for a hyperbola with vertices at (-2, -4) and (-2, 6) and foci at (-2, -5) and (-2, 7) can be written as:
(y - k)²/a² - (x - h)²/b² = 1
In this case, since the vertices and foci lie on the same vertical line, the equation becomes:
(y - (-1))²/10 - (x - (-2))²/c² = 1
where k is the y-coordinate of the center, h is the x-coordinate of the center, a is the distance from the center to the vertices, and c is the distance from the center to the foci.
Therefore, the standard form of the equation for this hyperbola is (y + 1)²/10 - (x + 2)²/c² = 1.