Ill write A for alpha and B for beta.
AB = c/a and A + B = -b/a
A^4 + B^4 = (A^2 + B^2)^2 - 2A^2B^2
= [(A + B)^2 - 2AB] ^2 - 2A^2B^2
Plugging in the values for A+B and AB we get
A^4 + B^4 = [(-b/a)^2 - 2c/a]^2 - 2(c/a)^2
= (b^2 / a^2 - 2c / a)^2 - 2c^2/a^2
= (b^2 - 2ac)^2 - 2c^2
---------------- -----
a^4 a^2
= (b^2 - 2ac)^2 - 2a^2c^2
-----------------------------
a^4
