Answer
$$v= u_1+2u_2-3u_3.$$
Work Step by Step
Suppose the linear combination $v= au_1+bu_2+cu_3$, then we have
$$(10,1,4)=a(2,3,5)+b(-1,2,4)+c(-2,2,3), \quad a,b,c\in R.$$
Which yields the following system of equations
\begin{align*}
2a-b-2c&=10\\
3a+2b+2c&=1\\
4a+4b+3c&=4.
\end{align*}
Solving the above system, we get the solution $a=1$, $b=2$, $c=-3$. Hence,
$$v= u_1+2u_2-3u_3.$$
