Answer :
Consider the given expression,
[tex]x^2+2x+2[/tex]Consider the properties,
[tex]\begin{gathered} i^2=-1 \\ i^3=-i \\ i^4=1 \end{gathered}[/tex]Let us check each expression, and the one which upon simplification gives the identical quadratic expression will be the correct choice.
Solve the first option,
[tex]\begin{gathered} (x-1+i)(x-1-i) \\ =(x-1)^2-(i)^2 \\ =(x^2-2x+1)-(-1) \\ =x^2-2x+1+1 \\ =x^2-2x+2 \end{gathered}[/tex]Since this is not identical to the given quadratic polynomial, first option is incorrect.
Solve the second option,
[tex]\begin{gathered} (x+2(x+1) \\ =x(x+1)+2(x+1) \\ =x^2+x+2x+2 \\ =x^2+3x+2 \end{gathered}[/tex]Since this is not identical to the given quadratic polynomial, second option is incorrect.
Solve the third option,
[tex]\begin{gathered} (x+1-i)(x+1+i) \\ =(x+1)^2-(i)^2 \\ =(x^2+2x+1)-(-1) \\ =x^2+2x+1+1 \\ =x^2+2x+2 \end{gathered}[/tex]Since this is identical to the given quadratic polynomial, third option is incorrect.
Thus, the third expression is equivalent to the given quadratic polynomial.