Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 2 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(12,6)= a(4,0)+b(0,3).$$
We get the system
\begin{align*}
4a&=12\\
3b&=6.
\end{align*}
By solving the above system we have the soluiton
$$a=3, \quad b=2.$$
Thus, the coordinate matrix of $x$ in $R^2$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 2 \end {array} \right].$$
