Answer
$$[x]_S=\left[ \begin {array}{ccc} -20\\ 32\\-4\\-5\end {array} \right].$$
Work Step by Step
Since we have
$$x= -2\left(4,0,7,3\right)+3\left(0,5,-1,-1\right)+4\left(-3,4,2,1\right)+(0,1,5,0)=\left(-20,32,-4,-5\right).$$
then we can write $x$ relative to the standard basis of $R^4$ as follows
$$x= \left(-20,32,-4,-5\right)=-20(1,0,0,0)+32 (0,1,0,0)-4(0,0,1,0)-5(0,0,0,1) .$$
Thus, he coordinates matrix of $x$ in $R^4$ relative to the
standard basis is
$$[x]_S=\left[ \begin {array}{ccc} -20\\ 32\\-4\\-5\end {array} \right].$$
