finding the value of k in an equation

Question

Find the value of k such that $f(x)=x^4-kx^3+kx^2+1$ is divisible by $d(x)=x+2$.

I tried using synthetic division for this problem and was able to get up to the part where k ends up being$(17+8k)$. My teacher said I am halfway done, What's next?

Answer 1

Hint: Find the roots of $f(x)$ (by Ruffini, for example) and set $k$ in such a way that one of the roots is at $x=-2$. Then you know that $f(x)$ will have a factorization which will include a factor $(x+2)$

Answer 2

A few hints to get you started:

If $f(x)$ is divisible by $x+2$ $\implies$ $f(-2) = 0$

Use this information to find values for $k$