Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} -5\\-11\\ 9 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(4,-2,9)= a(1,0,0)+b(0,1,0)+c(1,1,1).$$
We get the system
\begin{align*}
a+c&=4\\
b+ c&=-2\\
c&=9.
\end{align*}
By solving the above system we have the soluiton
$$a=-5, \quad b=-11, \quad c=9.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} -5\\-11\\ 9 \end {array} \right].$$