<s>[INST] If $x,y$ are distinct real numbers If $x,y$ are distinct real numbers, prove that $x+y=-2$ if and only if $(x+1)^2=(y+1)^2$
How would I prove this? do I use contrapositive, contradiction or direct proof?
 [/INST] $(x+1)^2=(y+1)^2 \iff (x+1)^2-(y+1)^2=0\iff (x+1-(y+1))(x+1+y+1)=0$
$ \iff x+1=y+1$ or $x+y+2=0$
Since we are excluding $x+1=y+1$ we have that the first is equivalent to $x+y=-2$.
</s>