Answer
The coordinate vector is;
$[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\end{bmatrix} = \begin{bmatrix}-6\\{ 2}\end{bmatrix}$
Work Step by Step
To find the coordinate vector $[x]_{\beta}$ of $x$ relative to the given basis $\beta$
Basis matrix
$\beta = \begin{bmatrix}1&5\\{ -2}&{ - 6}\end{bmatrix}$ and ${\bf{x}} = \begin{bmatrix}{ 4}\\0\end{bmatrix}$
finding $[x]_{\beta}$
$\beta {\left[ x \right]_\beta } = x$
$\begin{bmatrix}1&5\\{ -2}&{-6}\end{bmatrix}\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}=\begin{bmatrix}4\\0\end{bmatrix}$
forming augmented matrix and row reducing;
$\begin{bmatrix}1&5&4\\{ -2}&{-6}&0\end{bmatrix}\sim\begin{bmatrix}1&0&-6\\0&1&2\end{bmatrix}$
The coordinate vector is;
$[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\end{bmatrix} = \begin{bmatrix}-6\\{ 2}\end{bmatrix}$