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xy + x + y + 1 is an expression, not an equation. An expression cannot be solved.
You cannot. What you have in the question is an expression. An expression cannot be proven. You need an equation (or inequality).
The expression written in the question is the rational expression.
It is a quadratic expression and when factored is: (x+1)(x+1)
x does not equal -1, because if x is minus 1, x plus 1 is zero and division by zero is not defined.