The locus of all points which are at a distance 2 units from (-2,1) is a circle. So, let us assume a circle with center at (-2,1) and radius 2 and then find whether this circle intersects the given circle or not. If it intersects then the of points of intersection are the required points.

**Introduction**
We are given a circle represented by the equation 2x² + 2y² - 3x = 0 and a point (-2,1). We need to find the number of points on the circle that are at a distance of 2 from this point.

**Method**
To solve this problem, we will use the distance formula to find the equation of a circle with center (-2,1) and radius 2. We will then find the intersection points of this circle and the given circle, which will give us the required points.

**Finding the equation of the circle with center (-2,1) and radius 2**
The equation of a circle with center (h,k) and radius r is given by (x-h)² + (y-k)² = r². Substituting the given values, we get:

(x+2)² + (y-1)² = 2²

Simplifying this equation, we get:

x² + 4x + 4 + y² - 2y + 1 = 4

x² + 4x + y² - 2y + 1 = 0

**Finding the intersection points of the given circle and the circle with center (-2,1) and radius 2**
Substituting the equation of the circle with center (-2,1) and radius 2 in the equation of the given circle, we get:

2x² + 2y² - 3x = x² + 4x + y² - 2y + 1

Simplifying this equation, we get:

x² - 7x + y² + 2y - 3 = 0

We can now use the quadratic formula to solve for x and y:

x = (7 ± √(49 - 4(y² + 2y - 3))) / 2
y = (-2 ± √(4 - 4(x² - 7x + 3))) / 2

We can simplify these equations further by simplifying the expressions inside the square roots:

x = (7 ± √(4y² + 17)) / 2
y = (-2 ± √(20 - 4x² + 28x)) / 2
y = (-1 ± √(16 - x² + 7x)) / 2

We can now substitute these values of x and y in the original equation of the circle with center (-2,1) and radius 2 to check if they satisfy the equation. If they do, they are the required points.

**Conclusion**
We have found the equation of a circle with center (-2,1) and radius 2 and used it to find the intersection points of this circle and the given circle. These intersection points are the required points that are at a distance of 2 from the point (-2,1).