We are given the equation $x + \frac{1}{x} = \sqrt{2}$, and we need to find the value of $x^{80} + x^{76} + x^{72} + x^{65} + x^{64} + 4$.

Step 1: Analyze the given equation

We start with the equation: $x + \frac{1}{x} = \sqrt{2}$ Let's square both sides to get a simpler form involving powers of $x$: $\left( x + \frac{1}{x} \right)^2 = (\sqrt{2})^2$ $x^2 + 2 + \frac{1}{x^2} = 2$ Simplifying, we get: $x^2 + \frac{1}{x^2} = 0$ Thus: $x^2 = -\frac{1}{x^2}$ or equivalently: [ x^4