fa.functional analysis – Eigenvalues of operator

In the question here

the author asks for the eigenvalues of an operator

$$A = begin{pmatrix} x & -partial_x \ partial_x & -x end{pmatrix}.$$

Here I would like to ask if one can extend this idea to the operator

$$A = begin{pmatrix} x & -partial_x +c\ partial_x+c & -x end{pmatrix},$$

where $c$ is a real constant. It seems to me that this is a non-trivial change in the operator.