91Ó°ÊÓ

We'll rewrite the equation by combining the terms of the same variable, and completing the square to obtain the canonical sphere equation. Our equation: $$ x^{2}+y^{2}+z^{2}-6 x+6 y-8 z-2 = 0.$$ Rearrange terms, group corresponding variables and move the constant term to the right-hand side: $$ (x^2 - 6x) + (y^2 + 6y) + (z^2 - 8z) = 2.$$ Complete the squares by adding and immediately subtracting each square's term: $$ [(x^2 - 6x + 9) - 9] + [(y^2 + 6y + 9) - 9] + [(z^2 - 8z + 16) - 16] = 2.$$ Now, we can rewrite the equation in the canonical form for a sphere. Simplify the equation and complete the squares: $$ (x - 3)^2 - 9 + (y + 3)^2 - 9 + (z - 4)^2 - 16 = 2.$$ Move the added constants to the right-hand side, combine the constants on the right, and rewrite in a canonical form: $$ (x - 3)^2 + (y + 3)^2 + (z - 4)^2 = 32.$$

Now, our equation has the canonical form for a sphere: $$ (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 $$ Comparing it with our equation \((x - 3)^2 + (y + 3)^2 + (z - 4)^2 = 32\), we can find the center and the radius of the sphere: Center (a, b, c) = (3, -3, 4) and radius r = √32.

The geometric description of the set of points that satisfy the given equation is a sphere with center (3, -3, 4) and radius √32.