Trinomial

Algebra Level 5

Let a,ba,b be the roots of the square trinomial f(x)=x22x1f(x)=x^2-2x-1, and c,dc,d be the roots of the square trinomial g(x)=x23x1g(x)=x^2-3x-1.

The minimum value of g3(a)f(c)+g3(b)f(d)g^3(a)f(c)+g^3(b)f(d) can be expessed in the form m+nlm+n\sqrt{l}, where m,n,lm,n,l are integers and ll is a free-square number.

Find m+n+lm+n+l.

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