Trinomial

Algebra Level 5

Let \(a,b\) be the roots of the square trinomial \(f(x)=x^2-2x-1\), and \(c,d\) be the roots of the square trinomial \(g(x)=x^2-3x-1\).

The minimum value of \(g^3(a)f(c)+g^3(b)f(d)\) can be expessed in the form \(m+n\sqrt{l}\), where \(m,n,l\) are integers and \(l\) is a free-square number.