Answer
A point (-1,1,-1)
Work Step by Step
Given the surface:
\[
0=2 x-2 y+2 z+3+z^{2}+x^{2}+y^{2}
\]
Group the terms
\[
\begin{array}{l}
0=\left(1+2z+z^{2}\right)+\left(1-2y+y^{2}\right)+\left(1+2x+x^{2}\right) \\
\quad0=(1+z)^{2}+(1+x)^{2}+(-1+y)^{2}
\end{array}
\]
Since the sum of perfect squares is zero only if each individual terms are zero, the above equation is true if $-1=x, 1=y$ and $z=-1$. Thus, the given surface represents only a point (-1,1,-1).
