Solving polynominals equations (relationship of roots) 
The roots of $x^3-4x^2+x+6$ are $\alpha$, $\beta$, and $\omega$.
  Find (evaluate): 
  $$\frac{\alpha+\beta}{\omega}+\frac{\alpha+\omega}{\beta}+\frac{\beta+\omega}{\alpha}$$

So far I have found:
$$\alpha+\beta+\omega=\frac{-b}{a} = 4 \\
\alpha\beta+\beta\omega+\alpha\omega=\frac{c}{a} = 1 \\
\alpha×\beta×\omega=\frac{-d}{a} = -6$$
And evaluated the above fractions creating
$$\frac{\alpha^2\beta+\alpha\beta^2+\alpha^2\omega+\alpha\omega^2+\beta^2\omega+\beta\omega^2}{\alpha\beta\omega}$$ 
I don't know how to continue evaluating the question.
Note:
The answer I have been given is $-\dfrac{11}{3}$ 

Hint: We can write $$\frac{4-w}{w}+\frac{4-\beta}{\beta}+\frac{4-\alpha}{\alpha}$$ and this is $$4\left(\frac{\alpha\beta+\alpha w+w\beta}{\alpha \beta w}\right)-3$$ and this is $$-\frac{2}{3}\left(1-\beta w-\alpha w+\alpha w+\beta w\right)$$
This simplifies to $$-\frac{2}{3}-3=-\frac{11}{3}$$
