The factorization of $x^3 - x + 6$ is :

$x^3 - x + 6 = (x + 2)(x^2 - 2x + 3)$

To factorize the polynomial $x^3 - x + 6$, we'll use a combination of techniques, including the Rational Root Theorem and polynomial division.

### Step 1: Apply the Rational Root Theorem

The Rational Root Theorem states that any possible rational root of the polynomial $f(x) = x^3 - x + 6$ must be a factor of the constant term (here, 6) divided by a factor of the leading coefficient (here, 1).

The factors of 6 are: $\pm 1, \pm 2, \pm 3, \pm 6$

The factors of 1 are: $\pm 1$

So, the possible rational roots could be: $\pm 1, \pm 2, \pm 3, \pm 6$

### Step 2: Test possible rational roots

We will test these values by substituting them into the polynomial $f(x) = x^3 - x + 6$:

For $x = 1$:

$1^3 - 1 + 6 = 1 - 1 + 6 = 6 \quad (\text{not a root})$

For $x = -1$:

$(-1)^3 - (-1) + 6 = -1 + 1 + 6 = 6 \quad (\text{not a root})$

For $x = 2$:

$2^3 - 2 + 6 = 8 - 2 + 6 = 12 \quad (\text{not a root})$

For $x = -2$:

$(-2)^3 - (-2) + 6 = -8 + 2 + 6 = 0 \quad (\text{root found})$

Since $x = -2$ is a root, $x + 2$ is a factor of the polynomial.

### Step 3: Polynomial Division

We divide $x^3 - x + 6$ by $x + 2$ using synthetic or long division.

Using synthetic division with the root $-2$:

$\begin{array}{r|rrr} -2 & 1 & 0 & -1 & 6 \\ & & -2 & 4 & -6 \\ \hline & 1 & -2 & 3 & 0 \\ \end{array}$

So, the quotient is $x^2 - 2x + 3$.

### Step 4: Combine the factorization

Thus, we have:

$x^3 - x + 6 = (x + 2)(x^2 - 2x + 3)$

### Step 5: Check if quadratic can be factored further

To see if $x^2 - 2x + 3$ can be factored further, we'll check if it has real roots by finding the discriminant:

$\Delta = b^2 - 4ac = (-2)^2 - 4(1)(3) = 4 - 12 = -8$

Since the discriminant is negative ( $-8$ ), the quadratic $x^2 - 2x + 3$ has no real roots and cannot be factored further over the reals.

### Final Factorization

The final factorization of $x^3 - x + 6$ is:

$x^3 - x + 6 = (x + 2)(x^2 - 2x + 3)$

This is the most simplified factorization over the real numbers.