Find k such that f(x) = x^4-kx^3+kx^2+1 has the factor x+2

Answer 1

(x^4-kx^3+kx^2+1)/(x+2) = x^3 - (k+2)x^2 + (3k+4)x - (6k+8) with remainder 12k+17
So, you want 12k+17 = 0

Answer 2

or
by the Factor Theorem:
f(-2) = (-2)^4 - k(-2)^3 + k(-2)^2 + 1 = 0
16 + 8k + 4k + 1 = 0
12k + 17 = 0 , <------- oobleck's equation