What are the coordinates of the points where the graphs of

x^3-x^2+x+1 =x^3+x^2+x-1 

Subtract x^3, x   from each side

-x^2 + x + 1  =  x^2 - 1

-x^2 + 1  = x^2 - 1    rearange as

0  =  2x^2 - 2

2x^2 - 2  = 0     divide through by 2

x^2 - 1  = 0    factor

y = (1)^3 + (1)^2 + 1 - 1  =

(x + 1) ( x - 1)  = 0 

Set each factor to 0 and solve for x

And  x = 1   or x = -1

And using  x^3+x^2+x-1   

y = (1)^3 + (1)^2 + 1 - 1  =  2

y = (-1)^3 + (-1)^2 - 1 - 1  =  -2

So the solutions are  (1, 2)  and (-1. -2)