Use the remainder theorem: it says that if x - c is a linear factor of a polynomial f(x), then the remainder upon dividing f(x) by x - c is equal to f(c).
In this case,
f(p) = p⁴ - p² - p - k
and p - 1 is a factor of f(p). We have
f (1) = 1⁴ - 1² - 1 - k = -(k + 1)
Because p - 1 divides f(p), the remainder is 0, which means
-(k + 1) = 0
k + 1 = 0
k = -1