If alpha and beta are the zeroes of quadratic polynomial ax^2+bx+c , then find alpha^4+beta^4

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