Finding k for Polynomial with a Double Root

Consider the polynomial equation given by:

$$f(x) = 2x^4 - 3x^3 + 5x^2 - x + k$$

where $k$ is a constant. If the polynomial $f(x)$ has a double root at $x = 1$, determine the value of $k$ such that the conditions of the double root are satisfied. A double root implies that both $f(1) = 0$ and $f'(1) = 0$ must hold true. Find the suitable constant $k$ such that these conditions are fulfilled.