**Transformation of Equations with Parallel Axes**

When the axes are parallel, the given equation can be transformed using the following steps:

**Step 1: Determine the Center of the Hyperbola**

To determine the center of the hyperbola, we need to find the midpoint between the two given points on the axes. In this case, the midpoint can be obtained as follows:

Midpoint of x-axis: (0, 1)
Midpoint of y-axis: (-2, 0)

The center of the hyperbola is the midpoint of these two points, which is (-1, 0).

**Step 2: Translate the Equation to the New Coordinate System**

To translate the equation to the new coordinate system, we need to substitute the new coordinates (x', y') into the original equation. The new coordinates can be obtained using the formula:

x' = x - h
y' = y - k

where (h, k) represents the coordinates of the center of the hyperbola.

**Transformation of Equation a) x^2 - y^2 + 4x + 2y + 4 = 0**

Substituting the new coordinates into the equation, we have:

(x - 1)^2 - (y - 0)^2 + 4(x - 1) + 2(y - 0) + 4 = 0

Simplifying the equation, we get:

x^2 - 2x + 1 - (y^2 + 4x + 2y + 4) + 4 = 0
x^2 - 2x + 1 - y^2 - 4x - 2y - 4 + 4 = 0
x^2 - y^2 - 6x - 2y + 1 = 0

Therefore, the transformed equation is x^2 - y^2 - 6x - 2y + 1 = 0.

**Transformation of Equation b) y^2 - 4x - 2y - 7 = 0**

Substituting the new coordinates into the equation, we have:

(y - 0)^2 - 4(x - 1) - 2(y - 0) - 7 = 0

Simplifying the equation, we get:

y^2 - 4x + 4 - 2y - 7 = 0
y^2 - 2y - 4x - 3 = 0

Therefore, the transformed equation is y^2 - 2y - 4x - 3 = 0.

In conclusion, the transformed equations are:
a) x^2 - y^2 - 6x - 2y + 1 = 0
b) y^2 - 2y - 4x - 3 = 0

These transformations allow us to express the given equations in terms of the new coordinates (x', y') with the parallel axes passing through the point (-2,1).