What is the solution set of the given homogenous system?

Write the solution set of the given homogenous system in parametric vector form: \begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\ -4x_{1}-4x_{2}-8x_{3} &= 0\\ 0x_{1}-3x_{2}-3x_{3} &= 0\\ \end{align}

My attempt:

\begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\ -4x_{1}-4x_{2}-8x_{3} &= 0\\ 0x_{1}-3x_{2}-3x_{3} &= 0\\ \end{align}

Divided the second row by $-4$ and the third row by $-3$

\begin{align} x_{1}+x_{2}+2x_{3} &= 0\\ x_{1}+x_{2}+2x_{3} &= 0\\ 0x_{1}+x_{2}+x_{3} &= 0\\ \end{align}

Row $2$ and $3$ plus the first row multiplied by $-1$

\begin{align} x_{1}+x_{2}+2x_{3} &= 0\\ 0x_{1}+0x_{2}+0x_{3} &= 0\\ -x_{1}+0x_{2}-x_{3} &= 0\\ \end{align}

That looks fine so far except that the bottom left $0$ in the very last equation should be a $-1$. The middle equation tells you nothing, so you can eliminate it. You'll end up with a whole line of solutions (all multiples of a single vector).