We assumed that all polynomials of degree $\displaystyle \leq n$ can be factored into irreducibles. Since we wrote $\displaystyle p(x)=q(x)r(x)$ with $\displaystyle 1\leq \deg q, \deg r\leq n$, we can factor $\displaystyle q$ and $\displaystyle r$ into irreducibles, and therefore, $\displaystyle p$ can be factored into irreducibles.