Chapter 4 - Vector Spaces - 4.4 Spanning Sets and Linear Independence - 4.4 Exercises - Page 178: 27

$S$ does not span $P_2$.

Since $x^2+2=2(1)+1(x^2)$, then $S=\{1,x^2,x^2+2\}$ is not linearly independent. But $P_2$ has the dimension $3$, $S$ does not span $P_2$. $S$ spans only a subspace of $P_2$, precisely, it spans the subspace of all polynomials on the form $p(x)=a+bx^2$