To apply Newton's method for the function $f(x) = x^3 + x - 2$, we first need to find its derivative $f'(x)$.

1. Function: $f(x) = x^3 + x - 2$

2. Derivative: Differentiate $f(x)$ with respect to $x$.

   $f'(x) = 3x^2 + 1$

Newton's formula for finding the next approximation $x_{n+1}$ is:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Substitute $f(x_n)$ and $f'(x_n)$:

$x_{n+1} = x_n - \frac{x_n^3 + x_n - 2}{3x_n^2 + 1}$

Thus, the answer is:

$x_{n+1} = x_n - \frac{x_n^3 + x_n - 2}{3x_n^2 + 1}$

Would you like further details or have any questions?

Here are some related questions that could help deepen your understanding:

1. How does Newton's method work to approximate roots of functions?
2. Why is the derivative important in Newton's method?
3. What are the potential limitations of Newton's method?
4. Can Newton's method fail, and if so, under what conditions?
5. How would Newton’s method apply to a different function, such as $f(x) = e^x - 2x$?

Tip: Newton's method converges quickly when the initial guess is close to the actual root, but it may diverge if the initial guess is far or if the function's derivative is zero at any step.