Evaluating algebraic expression

$1. \ Expand \ and \ rearrange \ the \ expression \\ (1+x)^2+ \left ( 1- \dfrac{1}{x} \right )^2 \\ =(1+2x+x^2)+ \left (1- \dfrac{2}{x}+ \dfrac{1}{x^2} \right ) \\ =x^2+ \dfrac{1}{x^2} + 2 \left ( x- \dfrac{1}{x} \right ) +2 \\ = \left ( x- \dfrac{1}{x} \right )^2 + 2 \left ( x- \dfrac{1}{x} \right ) +4$

$2. \ Evaluate \ the \ value \ of \ x- \dfrac{1}{x} \\ \begin{aligned} x- \dfrac{1}{x} & = \dfrac{3+ \sqrt{5}}{2}- \dfrac{2}{3+\sqrt{5}} \\ & = \sqrt{5} \\ \end{aligned} \\ 3. \ Plug \ in \ x- \dfrac{1}{x}= \sqrt{5} \ to \ the \ expression \ above \ and \ we'll \ get \\ 9+2\sqrt{5}= \boxed{13.472} \quad (rounded \ to\ the \ nearest \ thousandth)$

Plug in and simplify.

Apply calculator for 13.472 as answer.