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.

Find $m+n+l$ .