a and b are the roots of x^2 - 2x - 2 = 0.

x^2 - 2x - 2 = 0

a = 2/2 + sqrt(4 + 8)/2

=> 1 + sqrt 12/2

=> 1 + sqrt 3

b = 1 - sqrt 3

a^2 + b^2 = (1 + sqrt 3)^2 + (1 - sqrt 3)^2

=> 1 + 3 + 2*sqrt 3 + 1 + 3 - 2*sqrt 3

=> 8

**The required value of a^2 + b^2 = 8**

There are two methods to determine the sum of the square of the roots of the given quadratic equation.

One of them is to use Viete's relations.

We know that the sum of the roots of the quadratic is the following:

a + b = -(-2)/1

a + b = 2

The product of the roots is:

a*b = -2/1

a*b = -2

The sum of the squares of the roots could be found using the formula:

(a + b)^2 = a^2 + 2ab + b^2

We'll subtract 2ab both sides:

(a + b)^2 - 2ab = a^2 + b^2

We'll replace a + b and a*b by its values:

(2)^2 - 2*(-2) = a^2 + b^2

4 + 4 = a^2 + b^2

a^2 + b^2 = 8

**The sum of the squares of the roots of the quadratic equation is a^2 + b^2 = 8.**