Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} -1\\ 1\\3\\2 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=\left(0,-20,7,15\right)= a\left(9,-3,15,4\right)+b\left(3,0,0,1\right)+c\left(0,-5,6,8\right)+d(3,-4,2,-3).$$
We get the system
\begin{align*}
9a+3b+3d&=0\\
-3a-5c-4d&=-20\\
15a+6c+2d&=7\\
4a+b+8c-3d&=15.
\end{align*}
By solving the above system we have the soluiton
$$a=-1, \quad b= 1, \quad c=3, \quad d=2.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} -1\\ 1\\3\\2 \end {array} \right].$$