Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} \frac{2}{5}\\-\frac{1}{4} \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(2,2)= a(5,0)+b(0,-8).$$
We get the system
\begin{align*}
5a&=2\\
-8b&=2.
\end{align*}
By solving the above system we have the soluiton
$$a=\frac{2}{5}, \quad b=-\frac{1}{4}.$$
Thus, the coordinate matrix of $x$ in $R^2$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} \frac{2}{5}\\-\frac{1}{4} \end {array} \right].$$
