Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 5\\ 1 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(-26,32)= a(-6,7)+b(4,-3).$$
We get the system
\begin{align*}
-6a+4b&=-26\\
7a-3b&=32.
\end{align*}
By solving the above system we have the soluiton
$$a=5, \quad b=1.$$
Thus, the coordinate matrix of $x$ in $R^2$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 5\\ 1 \end {array} \right].$$