Answer
$$v=2 u_1-u_2+3u_3.$$
Work Step by Step
Suppose the following linear combination
$$(3,0,-6)=a(1,-1,2)+b(2,4,-2)+c(1,2,-4), \quad a,b,c\in R.$$
Which yields the following system of equations
\begin{align*}
a+2b+c&=3\\
-a+4b+2c&=0\\
2a-2b-4c&=-6.
\end{align*}
Solving the above system, we get the solution $a=2$, $b=-1$, $c=3$. Hence,
$$v=2 u_1-u_2+3u_3.$$
