a1=((1+sqrt(5))/2)^(1/3)

a2=(abs(1-sqrt(5))/2)^(1/3)*sign(1-sqrt(5))

a=a1+a2

x=(a^2+a+1)

p=x^4+3*x^2-9*x+1

produces what is presumably wanted:

a =

         0.322185354626085

x =

          1.42598875736162

p =

         -1.59867276653908

alternatively the critical line could be

a2=(nthroot(1-sqrt(5))/2,3)

         

Without the absolute value and sign functions, or using the nthroot function, taking a negative number to the 1/3 power results in a complex cube root:

a1=((1+sqrt(5))/2)^(1/3)

a2=((1-sqrt(5))/2)^(1/3)

a=a1+a2

x=(a^2+a+1)

p=x^4+3*x^2-9*x+1

a =

           1.59988481774495 +     0.737680128975117i

x =

             4.615344275111 +      3.09808660637397i

p =

          -686.277403993008 +      727.277075162423i