[tex]x+\dfrac{1}{x} = 4 \Big|:x\\ \\ \Rightarrow 1 + \dfrac{1}{x^2} = \dfrac{4}{x}\Big|\cdot x^2 \\ \\ \Rightarrow x^2+1 = 4x \\ \\\text{Adun cele 3 egalitati:} \\\\ x+\dfrac{1}{x}+1+\dfrac{1}{x^2}+x^2+1= 4+\dfrac{4}{x}+4x\\ \\ \Rightarrow x^2+x+\dfrac{1}{x^2}+\dfrac{1}{x}+2 = 4+4\cdot\Big(\dfrac{1}{x}+x\Big)\\ \\ \Rightarrow x^2+x+\dfrac{1}{x^2}+\dfrac{1}{x} = 4-2 + 4\cdot 4 \\ \\ \Rightarrow x^2+x+\dfrac{1}{x^2}+\dfrac{1}{x} = \boxed{18}[/tex]
