Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 1\\ -1\\2 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(3,19,2)= a(8,11,0)+b(7,0,10)+c(1,4,6).$$
We get the system
\begin{align*}
8a+7b+c&=3\\
11a+4c&=19\\
10b++6c&=2.
\end{align*}
By solving the above system we have the soluiton
$$a=1, \quad b=-1, \quad c=2.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 1\\ -1\\2 \end {array} \right].$$