Let $V$ be the vector-space of the polynomial-functions $\in \mathbb{R}$ with $\deg \le n$ and $f \in \text{End}(V)$ with $c \in \mathbb{R}$:
$f\colon V \rightarrow V, p \mapsto f(p) := p'+cp$
How can I determine the transformation matrix of $f$ with respect to the monome-basis $B=(x^0,x,\dots,x^n)$?
In order to determine the matrix associated to $f$ with respect to $B$, you have to evaluate $f(x^i)$ for any element $x^i$ of $B$. More precisely, $$f(x^i)=ix^{i-1}+cx^i=q_i(x)$$
For any $i$, the polynomial $q_i(x)$ has coordinates $(0,\dots,\underbrace{i}_{i^{th}\text{ entry}},\underbrace{c}_{(i+1)^{th}\text{ entry}},\dots,0)=v_i$.
By definition, the $i^{th}$ column of the matrix associated to $f$ coincide with $v_i$. Hence the matrix associated to $f$ with respect to $B$ is $$ \begin{pmatrix} c & 1 & \dots & \dots & \dots & 0\\ 0 & c & 2 & \dots & \dots &0\\ 0 & 0 & c & \dots & \dots & 0\\ \dots &\dots & \dots &\dots & \dots &\dots\\ \dots &\dots & \dots &\dots & c &n\\ \dots &\dots & \dots &\dots & \dots &c \end{pmatrix} $$