Let $\lambda(A)$ denote the eigenvalues of any matrix $A$. First note that $\lambda(bA-dI) = b\lambda(A)-d$. So, we can ignore the scaling and shifting. Now, let us look at the slightly more general result (at least for the case $c \ge 0$ in your question).

If $\alpha_1\ge \cdots \ge \alpha_n$ are the eigenvalues of a matrix $A$, and $\beta_1 \ge \cdots \ge \beta_n$ are the eigenvalues of the matrix $A+xx^T$. Then, we have the following interlacing result:

\begin{equation*}
\beta_1 \ge \alpha_1 \ge \beta_2 \ge \alpha_2 \ge \cdots \ge \beta_n \ge \alpha_n.
\end{equation*}

I mention in passing, for a more general rank-1 perturbation, along with perhaps other useful specializations, you might benefit from skimming the paper "Eigenvalues of rank one perturbations of unstructured matrices" by A.C.M. Ran, M. Wojtylak, and also by chasing some of the references they provide for the simpler case that you are treating (namely, the rank-1 perturbation is: $c11^T$)---see also this paper.