To simplify this expression, we can factor the numerator and denominator:

[tex] \frac{ {x}^{3} + 1 }{ {x}^{2} - 1} = \frac{ (x+1)(x^2-x+1) }{ (x+1)(x-1) } [/tex]

Now we can cancel out the common factor of x + 1 from the numerator and denominator:

[tex] \frac{ (x+1)(x^2-x+1) }{ (x+1)(x-1) } = \frac{ x^2-x+1 }{ x-1 } [/tex]

Therefore, the simplified expression is:

[tex] \frac{ x^2-x+1 }{ x-1 } [/tex]