To answer the question, we need to re-arrange the given equation to be in the standard form. The given equation is y^2 - 12x + 4y - 32 = 0, which is a quadratic equation in two variables x and y.
Let's begin by rearranging the given equation such that the terms of y and x are separate:
y^2 + 4y + x - 12 = 32.
To continue, we need to complete the squares for the terms of x and y:
(y + 2)^2 - 4 + x - 12 = 32.
Simplified, this becomes:
(y + 2)^2 + x - 16 = 0.
This can be re-written as:
x = - (y + 2)^2 + 16.
Finally, let's rewrite this formula in standard form:
(x - 16) = - (y - (-2))^2.
This transformation concludes the process of converting the given equation to standard form. The standard form of the given equation is (x - 16) = - (y - (-2))^2.