$(x + 2)$ and $(x + 3)$ are the factors of $x^3 + 4x^2 + x - 6$. The third factor of the given polynomial is :

Given,

$(x + 2)$ and $(x + 3)$ are the factors of $x^3 + 4x^2 + x - 6$.

⇒ $(x + 2)(x + 3) = x^2 + 5x + 6$

Thus, $x^2 + 5x + 6$ is also the factor of $x^3 + 4x^2 + x - 6$.

$$\begin{array}{l} \phantom{x^2 + 5x + 6\big)\quad }\phantom{x^3 +} x - 1 \\ x^2 + 5x + 6\overline{\smash{\big)}\quad x^3 + 4x^2 + x - 6} \\ \phantom{+++++()}\underline{\underset{-}{+}x^3 \underset{-}{+}5x^2 \underset{-}{+}6x} \\ \phantom{+++++())}\phantom{x^3 +} -x^2 - 5x - 6 \\ \phantom{+++++())}\phantom{x^3 +} \underline{\underset{+}{-}x^2 \underset{+}{-}5x \underset{+}{-}6} \\ \phantom{x^2 + 5x + 6\big)\quad }\phantom{x^3 + 4x^2 +} \times \end{array}$$

Thus,

$$\begin{aligned}x^3 + 4x^2 + x - 6 = (x^2 + 5x + 6)(x - 1) \\[1em] = (x + 2)(x + 3)(x - 1).\end{aligned}$$

Thus, the third factor of the given polynomial is $(x - 1)$.

Hence, option 1 is the correct option.