I tried setting up the systems of equations like $4a+8b+4c = 0$, etc. and solving for the variables. I used rref on my calculator and it gave me a solution where $c = 0$ and $a$ and $b$ were both non-zero numbers. However this solution was wrong. Can someone tell me where to start with this question?
HINT
Solve the system
$$\begin{bmatrix} 4 & 8 & 4\\ -5 & -10 & -5\\ -3 & -1 & -2\end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$
if the only solution is $a=b=c=0$ then the given vectors are linearly independent.
Your solution with $c=0$ seems to be not correct since the first two vectors are linearly independent.