**Solution:**

To find the value of 'a' for which the sum of the squares of the roots of the given equation assumes the least value, we need to follow the steps below:

**Step 1: Find the Roots of the Equation:**

The given equation is a quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = -(a-2), and c = -(a+1).

Using the quadratic formula, the roots of the equation can be found as follows:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values of a, b, and c, we get:

x = (-(a-2) ± √((a-2)² - 4(a)(-(a+1)))) / 2(1)

Simplifying further:

x = (2-a ± √(a² - 4a + 4 + 4a + 4)) / 2

x = (2-a ± √(a² + 4)) / 2

Hence, the roots of the equation are:

x₁ = (2-a + √(a² + 4)) / 2

x₂ = (2-a - √(a² + 4)) / 2

**Step 2: Find the Sum of the Squares of the Roots:**

The sum of the squares of the roots can be found by squaring each root and then adding them together:

Sum of squares of roots = x₁² + x₂²

Substituting the values of x₁ and x₂, we get:

Sum of squares of roots = [(2-a + √(a² + 4)) / 2]² + [(2-a - √(a² + 4)) / 2]²

Simplifying further:

Sum of squares of roots = [(2-a)² + (a² + 4) + 2(2-a)√(a² + 4)] / 4 + [(2-a)² + (a² + 4) - 2(2-a)√(a² + 4)] / 4

Sum of squares of roots = [(2-a)² + (a² + 4)] / 2

Sum of squares of roots = (4 - 4a + a² + a² + 4) / 2

Sum of squares of roots = (2a² - 4a + 8) / 2

Sum of squares of roots = a² - 2a + 4

**Step 3: Find the Least Value of the Sum of Squares of Roots:**

We have the sum of squares of roots as a quadratic expression in terms of 'a':

Sum of squares of roots = a² - 2a + 4

To find the least value of this expression, we can analyze its vertex. The vertex of a quadratic equation in the form of ax² + bx + c is given by (-b/2a, f(-b/2a)), where f(x) is the quadratic function.

In our case, a = 1, b = -2, and c = 4. Therefore, the vertex will be:

x = -(-2