Since we're considering a subset of the vector space `P_2` , with its usual definitions of addition and scalar multiplication, we only need to check that `S` is nonempty and closed under both operations. `S` is nonempty because `0+t^2 in S.` However, `S` is not closed under addtion because
`(0+t^2)+(0+t^2)=0+2t^2 notin S.`
There are other, equally fast ways to show `S` isn't a vector space. You can realize that `S` doesn't contain the zero vector because each of its elements has the term `1t^2.`
You can say that `S` isn't closed under scalar multiplication because `0(0+t^2)=0+0t^2,` which isn't an element of `S`.
However you show it, `S` is not a subspace of `P_2.`