Consider the set {1+x^2, 4-2x, 1+x+x^2} of three vectors in the Vector Space P_2. Does S span P_2?
I proved that it was closed under vector addition and multiplication is that enough or is there more?
The idea here is to test and see if these functions are linearly independent. If they are linearly independent, they also span the set.
If we put the coefficients of the polynomials in a matrix and take its determinant, we see that
$\displaystyle \begin{vmatrix}1 & 0 & 1\\ 4 & -2 & 0\\1 & 1 & 1\end{vmatrix}=\begin{vmatrix}-2 & 0\\ 1 & 1\end{vmatrix}+\begin{vmatrix}4 & -2\\1 & 1\end{vmatrix}=-2+6=4\neq 0$.
Therefore, the polynomials in set $\displaystyle S$ are linearly independent and span $\displaystyle P_2$.
I suppose you meant to ask whether $\displaystyle Span(S)=P_2$, and since $\displaystyle \dim P_2=3$ , three vectors span it iff they're a basis iff they're linearly independent, so you've to check whether $\displaystyle a_1(1+x^2)+a_2(4-2x)+a_3(1+x+x^2)=0 \Longleftrightarrow a_1=a_2=a_3=0$
Tonio