For example determine the basis for span S={$\displaystyle t-2,2t-1,4t-2,t^2-t+1,t^2+2t+1$}

how would I do it if I do/do not want the basis to be a subset of S?

I know that if I want it to be a subset then I put the matrix $\displaystyle \left[ \begin{array}{cccc}

0 & 0 & 1 & 1 \\

1 & 2 & -1 & 2 \\

-1 & -2 & 1 & 1

\end{array} \right]$ into reduced row echelon forum which gives $\displaystyle \left[ \begin{array}{cccc} 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$

and I don't know where to go from here.