Given that (x-1) is a factor of f(x) = ax⁴ + x³ - 12x² - x + 2, by the Factor Theorem, f(1) = 0.

Given that $(x-1)$ is a factor of $f(x) = ax^4 + x^3 - 12x^2 - x + 2$, by the Factor Theorem, $f(1) = 0$.
Substitute $x=1$ into $f(x)$:

The value of the constant $a$ is $\boxed{10}$.

Now, substitute $a=10$ into $f(x)$, so $f(x) = 10x^4 + x^3 - 12x^2 - x + 2$.
Substitute $x=-1$ into $f(x)$:

This verifies that $f(-1) = 0$.

1. (ii)

Let the roots of the equation $x^2 + (k+1)x + k = 0$ be $\alpha$ and $2\alpha$.

For a quadratic equation $Ax^2 + Bx + C = 0$, the sum of roots is $-\frac{B}{A}$ and the product of roots is $\frac{C}{A}$.
In this equation, $A=1$, $B=(k+1)$, $C=k$.

Sum of roots:

Product of roots:

From equation (1), express $\alpha$ in terms of $k$:

Substitute this expression for $\alpha$ into equation (2):

Factor the quadratic equation:

This gives two possible values for $k$:

The values of the constant $k$ are $\boxed{\frac{1}{2} \text{ and } 2}$.