Write the system in augmented-matrix form:
[tex]c_1(1,0,1,0)+c_2(1,0,-2,1)+c_3(2,0,1,2)=(1,-2,2,3)[/tex]
[tex]\iff\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\1&-2&1&2\\0&1&2&3\end{array}\right][/tex]
Row reduce this matrix:
[tex]\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&-3&-1&1\\0&1&2&3\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&5&10\\0&1&2&3\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&1&2\\0&1&2&3\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&1&2\\0&1&0&-1\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc|c}1&0&0&-2\\0&0&0&-2\\0&0&1&2\\0&1&0&-1\end{array}\right][/tex]
This matrix tells us that [tex]c_1=-2[/tex], [tex]c_2=-1[/tex], and [tex]c_3=2[/tex], but clearly [tex]0c_1+0c_2+0c_3=0\neq-2[/tex], so there is no solution.