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 $-\dfrac{11}{3}$ 

That follows from your results,  since we get:  $\dfrac{4-\omega}{\omega}+\dfrac{4-\beta}{\beta}+\dfrac{4-\alpha}{\alpha}=\dfrac{4(\omega\beta+\omega \alpha+\beta\alpha)-3\omega\beta\alpha}{\omega \beta \alpha}=\dfrac{4+18}{-6}=-\dfrac{11}3$.
