If $$a^{2} + b^{2}+ c^{2} = 2(a - b - c) - 3$$, then the value of (a - b + c) is

$$a^2+b^2+c^2=2\left(a-b-c\right)-3$$

$$a^2+b^2+c^2=2a-2b-2c-3$$

$$a^2+b^2+c^2-2a+2b+2c+3=0$$

$$a^2-2a+1+b^2+2b+1+c^2+2c+1=0$$

$$\left(a-1\right)^2+\left(b+1\right)^2+\left(c+1\right)^2=0$$

Sum of squares are zero so each term is zero

$$\left(a-1\right)^2=0$$,  $$\left(b+1\right)^2=0$$,  $$\left(c+1\right)^2=0$$

$$a-1=0$$,     $$b+1=0$$,    $$c+1=0$$

$$=$$>  a = 1, b = -1, c =-1

Therefore a-b+c = 1-(-1)-1= 1+1-1= 1