For polynomials $f(x), p(x)$ in a polynomial ring $K[X]$, $K$ a field, where $p(x)|f(x)$ does it then imply that $\frac{f(x)}{p(x)} \in K[X]$ aswel?

Question

I'm doing a problem about extension fields $K \subset L$, $L$ being algebraic and have that deg(Irr$(\alpha, K)$) = 2 for an $\alpha \in L$. This means Irr$(\alpha, K) = (X-\alpha)(X-\beta)$ for some $\beta$. My idea was that $(X-\beta) = \frac{\text{Irr}(\alpha, K)}{X-\alpha} \in L[X]$ $\Rightarrow$ $\beta \in L$ but i'm suddenly not sure about the fact about the polynomials. It seems true obviously, but is this true?