We have
$$x^2 + \frac{1}{x^2} + 1 = \left(x + \frac{1}{x}\right)^2 - 1 = \left(x + \frac{1}{x} + 1\right)\left(x + \frac{1}{x} - 1\right).$$
Since $x + \frac{1}{x} + 1$ is a positive integer, $x + \frac{1}{x} - 1$ is an integer and so is $x^2 + \frac{1}{x^2} + 1$. Since $x^2 + \frac{1}{x^2} + 1 \ge 1$ we conclude that $x^2 + \frac{1}{x^2} + 1$ is a positive integer divisible by $x + \frac{1}{x} + 1$.