To factor the expression x^3 + x + 2, we can use the Rational Root Theorem to find any possible rational roots. The Rational Root Theorem states that if a polynomial has a rational root p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p must be a factor of the constant term and q must be a factor of the leading coefficient.

In this case, the constant term is 2 and the leading coefficient is 1. The factors of 2 are ±1 and ±2, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1 and ±2.

We can now test these possible roots by substituting them into the polynomial and checking if the result is equal to zero.

For x = 1:
(1)^3 + (1) + 2 = 1 + 1 + 2 = 4 ≠ 0

For x = -1:
(-1)^3 + (-1) + 2 = -1 - 1 + 2 = 0

Therefore, x = -1 is a root of the polynomial. By using synthetic division or long division, we can divide the polynomial by (x + 1) to find the remaining factors.

(x^3 + x + 2) ÷ (x + 1) = x^2 - x + 2

The remaining quadratic factor x^2 - x + 2 cannot be factored further using rational numbers. Therefore, the factored form of x^3 + x + 2 is (x + 1)(x^2 - x + 2).