Answer
${\bf{z}} = \left\langle {0, - 2, - 3} \right\rangle $
Work Step by Step
$$\eqalign{
& {\text{Let the vectors:}} \cr
& {\bf{u}} = \left\langle {1,2,3} \right\rangle ,{\text{ }}{\bf{v}} = \left\langle {2,2, - 1} \right\rangle ,{\text{ }}{\bf{w}} = \left\langle {4,0, - 4} \right\rangle \cr
& \cr
& {\text{Find the vector }}{\bf{z}}, \cr
& {\text{2}}{\bf{u}} + {\bf{v}} - {\bf{w}} + 3{\bf{z}} = 0 \cr
& {\text{Substituting the given vectors}} \cr
& {\text{2}}\left\langle {1,2,3} \right\rangle + \left\langle {2,2, - 1} \right\rangle - \left\langle {4,0, - 4} \right\rangle + 3{\bf{z}} = 0 \cr
& {\text{Solve and simplify}} \cr
& \left\langle {2,4,6} \right\rangle + \left\langle {2,2, - 1} \right\rangle - \left\langle {4,0, - 4} \right\rangle + 3{\bf{z}} = 0 \cr
& \left\langle {2 + 2 - 4,4 + 2 - 0,6 - 1 -(- 4)} \right\rangle + 3{\bf{z}} = 0 \cr
& \left\langle {0,6,9} \right\rangle + 3{\bf{z}} = 0 \cr
& 3{\bf{z}} = - \left\langle {0,6,9} \right\rangle \cr
& {\bf{z}} = - \frac{1}{3}\left\langle {0,6,9} \right\rangle \cr
& {\bf{z}} = \left\langle {0, - 2, - 3} \right\rangle \cr} $$
