Prove $\displaystyle \frac{n^3+2n}{n^4+3n^2+1}$ is a fraction in its lowest terms

Rewriting

$\displaystyle \frac{n(n^2+2)}{(n^2+1)^2+n^2}$

The numerator is a product, so we shall consider them separately

$\displaystyle \frac{n^2+2}{(n^2+1)^2+n^2}$

Let $\displaystyle z=n^2+1$

$\displaystyle \frac{z+1}{z^2+z-1}$. Numerator and denominator share no common factors

Consider

$\displaystyle \frac{n}{(n^2+1)^2+n^2}$

This also has no common factors

Hence

$\displaystyle \frac{n^3+2n}{n^4+3n^2+1}$

has no common factors

QED

Is this proof valid?