Proving a subset is a vector subspace using Subspace criterion

Question

Given $$V =\ {f :\mathbb R\to \mathbb R \mid f(x)=a+bx+cx^2\mbox{ where }a, b, c \in \mathbb R}$$ and $$W = \{f :\mathbb R\to \mathbb R \mid f(x)=α+βx^2\mbox{ where }α, β \in \mathbb R\}$$ how do I show, using the Subspace criterion, that the subset $W \subseteq V$ is a vector subspace of $V$?

Answer 1

Well, let $f=a+bx^2$ and $g=c+dx^2$ be elements of $W$.

Then $f+g = (a+bx^2) + (c+dx^2) = (a+c) + (b+d)x^2$ lies in $W$.

Moreover, for each scalar $t$, we have that

$tf = t(a+bx^2) = (ta) + (tb)x^2$ lies in $W$ too.