lg^2 (x + 1) - lg (x + 1) lg (x - 1) lg^2 (x - 1)

= (lg(x + 1))^2 - lg(x + 1) lg(x - 1) (lg(x - 1))^2

= (lg(x + 1))^2 - (lg(x + 1))(lg(x - 1))(lg(x - 1))^2

= (lg(x + 1))(lg(x + 1) - (lg(x - 1))^2

So the simplified expression is lg(x + 1)(lg(x + 1) - (lg(x - 1))^2.