How to solve this big O notation of a fraction
The idea is that $x^4 + x^2 + 1 = \Theta(x^4)$, and similarly $x^4 + 1 = \Theta(x^4)$. So the first expression is $\Theta(x^4)/\Theta(x^4) = \Theta(1)$. The second expression can be estimated similarly.

This sort of calculation shows the advantage of asymptotic notation - irrelevant details disappear and calculations become manageable.