Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 1\\0\\1 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=\left(21, -5, 43, 14\right)= a\left(9, -3, 15, 4\right)+b\left(-3, 0, 0,-1\right)+c\left(0, -5, 6, 8\right)+d(-3, 4,-2, 3).$$
We get the system
\begin{align*}
9a-3b-3d&=21\\
-3a-5c+4d&=-5\\
15a+6c-2d&=43\\
4a-b+8c+3d&=14.
\end{align*}
By solving the above system we have the soluiton
$$a=3, \quad b= 1, \quad c=0, \quad d=1.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 3\\ 1\\0\\1 \end {array} \right].$$
