  In the present paper, we prove that all the quotient modules in $H^2(\mathbb
D^2)$, associated to the finitely generated submodules containing a
distinguished homogenous polynomial, are essentially normal, which is the first
result on the essential normality of non-algebraic quotient modules in
$H^2(\mathbb D^2)$. Moreover, we obtain the equivalence of the essential
normality of a quotient module and the Hilbert-Schmidtness of its associated
submodule in $H^2(\mathbb D^2)$, in the case that the submodule contains a
distinguished homogenous polynomial. As an application, we prove that each
finitely generated submodule containing a polynomial is Hilbert-Schmidt, which
partially gives an affirmative answer to the conjecture of Yang \cite{Ya3}.
