$${\textbf{Step -1: Considering option A.}}$$
$${\text{We have given,}}$$ $$x +1 = 0$$
$${\text{Putting }}x = - 1{\text{ we get,}}$$
$$ - 1 +1 = 0$$
$$ \Rightarrow 0 = 0$$ $${\text{Which is possible}}{\text{.}}$$
$${\text{Here, L}}{\text{.H}}{\text{.S}} = {\text{R}}{\text{.H}}{\text{.S }}$$
$${\text{So, - 1 is a solution of }}x +1 = 0.$$
$${\textbf{Step -2: Considering option B.}}$$
$${\text{We have given,}}$$ $$3x + 4 = 1$$
$${\text{Putting }}x = - 1{\text{ we get,}}$$
$$3\left( { - 1} \right) + 4 = 1$$
$$ \Rightarrow - 3 + 4 = 1$$
$$ \Rightarrow 1 = 1$$ $${\text{Which is possible}}{\text{.}}$$
$${\text{Here, L}}{\text{.H}}{\text{.S }} = {\text{ R}}{\text{.H}}{\text{.S }}$$
$${\text{So, - 1 is a solution of 3}}x + 4 = 1.$$
$${\textbf{Step -3: Considering option C.}}$$
$${\text{We have given,}}$$ $$5x + 7 = 2$$
$${\text{Putting }}x = - 1{\text{ we get,}}$$
$$5\left( { - 1 } \right) + 7 = 2$$
$$ \Rightarrow -5+ 7 = 2$$
$$ \Rightarrow 2 = 2$$ $${\text{Which is possible}}{\text{.}}$$
$${\text{Here, L}}{\text{.H}}{\text{.S = R}}{\text{.H}}{\text{.S }}$$
$${\text{So, - 1 is a solution of }}5x+7 =2.$$
$${\textbf{Step -4: Considering option D.}}$$
$${\text{We have given,}}$$ $$x -1 = 2$$
$${\text{Putting }}x = - 1{\text{ we get,}}$$
$$-1-1= 2$$
$$ \Rightarrow - 2 = 2$$ $${\text{Which is not possible}}{\text{.}}$$
$${\text{Here, L}}{\text{.H}}{\text{.S}} \ne {\text{R}}{\text{.H}}{\text{.S }}$$
$${\text{So, - 1 is not a solution of }}x -1 = 2.$$
$${\textbf{ Hence, option D. }}\mathbf{ x-1 =2}{\textbf{ is correct answer.}}$$