$(x – 1)$ is a factor of the polynomial $x^3 + 2x^2 – x – k$. The value of $k$ is:

Given,

$(x – 1)$ is a factor of $x^3 + 2x^2 – x – k$.

∴ On putting $x = 1$, the value of the polynomial is zero.

$$\begin{aligned}\Rightarrow (1)^3 + 2(1)^2 - (1) - k = 0 \\[1em] \Rightarrow 1 + 2 - 1 - k = 0 \\[1em] \Rightarrow 2 - k = 0 \\[1em] \Rightarrow k = 2.\end{aligned}$$

Hence, option 4 is the correct option.