Answer
$$[x]_S=\left[ \begin {array}{ccc} -\frac{9}{2}\\ 29\\11\end {array} \right].$$
Work Step by Step
Since we have
$$x= 2\left(\frac{3}{4},\frac{5}{2},\frac{3}{2}\right)+0\left(3,4,\frac{7}{2}\right)+4\left(-\frac{3}{2},6,2\right)=\left(-\frac{9}{2},29,11\right).$$
then we can write $x$ relative to the standard basis of $R^3$ as follows
$$x= \left(-\frac{9}{2},29,11\right)=-\frac{9}{2}(1,0,0)+29(0,1,0)+11(0,0,1) .$$
Thus, he coordinates matrix of $x$ in $R^3$ relative to the
standard basis is
$$[x]_S=\left[ \begin {array}{ccc} -\frac{9}{2}\\ 29\\11\end {array} \right].$$
