The question is
Does there exist a vector space $V$ over the real numbers $\mathbb{R}$ and three vectors $\vec{v}_1,\vec{v}_2, \vec{v}_3$ in $V$ such that the 4 vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4 = \vec{v}_1 − 2\vec{v}_2 + \vec{v}_3$ are linearly independent? Justify your answer.
Solving for $$a \cdot \vec{v}_1 + b\cdot\vec{v}_2+ c\cdot\vec{v}_3 + d\cdot\vec{v}_4 = \vec{0}$$ gives $a = c, b = -2a$ for the system to have 0 for each coefficient but I'm not sure where to go from there, mainly because I don't really understand what the question is looking for.
You forgot $d=-a$ and to say that $a$ is any real number (you only needed to pick one though, since would rescale both sides and $a*0=0$), but otherwise you are done. Your linear dependence works for all possible choices in the question. All $V$ and all $v_{1,2,3}$ in that $V$.