State true or false --> x = Θ(n⁴), y = Θ(n²) therefore x/y = Θ(n²)
It's true.  There are positive constants $c_1$, $c_2$, $c_3$ and $c_4$ such that, for all large enough $n$,
$$c_1n^4 \leq x\leq c_2 n^4 \quad\text{and}\quad c_3n^2\leq y\leq c_4n^2\,.$$  Therefore,
$$\frac{c_1}{c_4}n^2 \leq \frac{x}{y}\leq \frac{c_2}{c_3} n^2\,,$$
i.e., $x/y=\Theta(n^2)$.

By the way, in your argument, you say that $x$ and $y$ are polynomials. That's not necessarily true&nbsp;&ndash; for example, we could have $x=n^4+\log n$.