Sorry, I forgot about this question over the weekend, as I got pretty busy. Perhaps you already have decoded the solution, but based on what you wrote, I will give you my interpretation of what is going on.
From the solution you present, it seems that p is supposed to be a negative real number, and q perhaps represents the absolute value of its cube roots. We name the three values of the cube roots as a, b, and c.
Now, the symbol ω represents the primary cube root of unity. That means it is the complex number
ω = ei2π/3
and note that ω3 = 1. It is a basic theorem in complex analysis that we can express the nth roots of a complex number in terms of combinations of the nth roots of unity, which are similarly defined.
Note also that
ω2 = ei4π/3
is a complex number, which, when cubed, gives us 1, and similarly,
ω3 = 1.
In other words, both ω and ω2 are cube roots of unity. Since 1 itself is also a cube root of unity (unity here just means 1 of course), we can now represent the cube roots of any real number p in terms of these roots of unity. Letting q = p1/3, we would have
a = q
b = qω
c = qω2.
If p < 0, we can of course choose q = -p1/3, or what is the same, -q = p1/3 and it seems that is what your question asked. In any case, setting up the expression we are asked to evaluate, we now get
(xa + yb + zc)/(xb + yc + za) = -(xq + yqω + zqω2)/-(xqω + yqω2 + zq)
= (x + yω + zω2)/(xω + yω2 + z)
If we now multiply numerator and denominator by ω2, and use the fact that ω4 = ω (which you can verify), we will have
ω2(x + yω + zω2)/(x + yω + zω2) = ω2