$F[x]/\langle f(x)\rangle$ vs. $F[x]/\langle g(x)\rangle$ when $\deg f(x)<\deg g(x)$

Question

Let $F$ be a field, $f(x),~g(x)\in F[x]$ be irreducible and $\deg f(x)<\deg g(x)$. Then is $F[x]/\langle f(x)\rangle$ strictly contained in $F[x]/\langle g(x)\rangle$? For example, does $\Bbb R\subsetneq \Bbb R[x]/\langle x^2+1\rangle\subsetneq \Bbb R[x]/\langle x^4+1\rangle\subsetneq \Bbb R[x]/\langle x^6+1\rangle\subsetneq\cdots\cdots\subsetneq \Bbb R[x]/\langle x^{2k}+1\rangle$?