Answer
$${\text{Sphere centered at }}\left( {3, - 3,4} \right){\text{ with radius }}6$$
Work Step by Step
$$\eqalign{
& {x^2} + {y^2} + {z^2} - 6x + 6y - 8z - 2 = 0 \cr
& {\text{Group terms}} \cr
& \left( {{x^2} - 6x} \right) + \left( {{y^2} + 6y} \right) + \left( {{z^2} - 8z} \right) = 2 \cr
& {\text{Complete the square}} \cr
& \left( {{x^2} - 6x + 9} \right) + \left( {{y^2} + 6y + 9} \right) + \left( {{z^2} - 8z + 16} \right) = 2 + 9 + 9 + 16 \cr
& {\left( {x - 3} \right)^2} + {\left( {y + 3} \right)^2} + {\left( {z - 4} \right)^2} = {\left( 6 \right)^2} \cr
& {\text{The equation is in the form }}{\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} + {\left( {z - c} \right)^2} = {r^2} \cr
& {\text{Represents a sphere centered at }}\left( {a,b,c} \right){\text{ with radius }}r,{\text{ then}} \cr
& {\left( {x - 3} \right)^2} + {\left( {y + 3} \right)^2} + {\left( {z - 4} \right)^2} = {\left( 6 \right)^2} \cr
& {\text{Sphere centered at }}\left( {3, - 3,4} \right){\text{ with radius }}6 \cr} $$
