Determine the value of K such that when:
f(x) = x^4 + kx^2 - 3x - 5
is divided by (x-1), the is remainder -10
Reason being, because of the Polynomial Remainder Theorem, the remainder $\displaystyle r$ of a polynomial $\displaystyle P(x)$ when divided by $\displaystyle (x-a)$ is such that $\displaystyle r = P(a)$. Proof of why this happens in the link. It's not definetely NOT complicated, but it's senseless to just glue it here.
As for your problem, it's a pretty much direct application. You want the remainder of that division to be -10, so $\displaystyle f(1) = 10$. This leaves you with an equation with one unknown. Now just solve for k.