To find the area of the shaded portion, we need to subtract the area of the triangle from the area of the square.

1. Area of the Square:
The area of a square is given by the formula A = side², where A is the area and side is the length of one side of the square.
In this case, the area of the square is given as 4x² - 2x - 6 square units.

2. Area of the Triangle:
The area of a triangle is given by the formula A = (base * height) / 2, where A is the area, base is the length of the base of the triangle, and height is the height of the triangle.
In this case, the area of the triangle is given as x² - 4x + 5 square units.

3. Finding the Side Length of the Square:
Since the area of the square is given as 4x² - 2x - 6 square units, we can equate it to the formula A = side² and solve for the side length.
4x² - 2x - 6 = side²
Taking the square root of both sides, we get:
side = √(4x² - 2x - 6)

4. Calculating the Area of the Square:
Now that we know the side length of the square, we can calculate its area.
Area of the square = side² = (√(4x² - 2x - 6))² = 4x² - 2x - 6 square units.

5. Calculating the Area of the Triangle:
The area of the triangle is given as x² - 4x + 5 square units.

6. Finding the Area of the Shaded Portion:
To find the area of the shaded portion, we subtract the area of the triangle from the area of the square.
Area of the shaded portion = Area of the square - Area of the triangle
Area of the shaded portion = (4x² - 2x - 6) - (x² - 4x + 5)
Simplifying, we get:
Area of the shaded portion = 4x² - 2x - 6 - x² + 4x - 5
Combining like terms, we get:
Area of the shaded portion = 3x² + 2x - 11 square units.

Therefore, the area of the shaded portion is 3x² + 2x - 11 square units.