A polynomial, $P$, is $x^3 - x^2 - x + k$, where $k$ is a constant.

1. Apply the Remainder Theorem by substituting $x = 2$ into the polynomial and equating to the given remainder.

Let $P(x) = x^3 - x^2 - x + k$.
By the Remainder Theorem, $P(2) = 3$.
$2^3 - 2^2 - 2 + k = 3$

2. Solve the equation for $k$.

$8 - 4 - 2 + k = 3$
$2 + k = 3$
$k = 1$

1. Set up the identity using the given factor and an unknown linear factor

Since $x^2 + ax + 1$ is a quadratic factor of the cubic polynomial $P(x) = x^3 - x^2 - x - 2$, the other factor must be linear. Let the linear factor be $(x + c)$.
$x^3 - x^2 - x - 2 = (x^2 + ax + 1)(x + c)$

2. Determine the constant $c$ by comparing the constant terms

Comparing the constant terms on both sides:
$-2 = 1 \times c$
$c = -2$

3. Determine $a$ by comparing the coefficients of $x^2$

Substitute $c = -2$ into the identity:
$x^3 - x^2 - x - 2 = (x^2 + ax + 1)(x - 2)$
Comparing the coefficients of $x^2$:
$-1 = -2 + a$
$a = 1$