Mathy Person

I am having some trouble with this problem: Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}$, (precalculus, roots of unity). Does anyone have any hints? I tried rearranging it to cancel some terms out in the fractions but it ended up becoming a large, messy fraction.

Can anyone help me with this problem? Roots of unity. "Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
\[\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}"

I got the answers $k = 4, 9$ for this problem: Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector ${v}$ such that
$\begin{pmatrix} 2 & 12 \\ 2 & -3 \end{pmatrix} {v} = k {v}$, but the way I obtained these answers seem a bit too easy. Is $k=4, 9$ correct?