Subspace Advance Linear Algebra, Cooperstein

Question

In Exercises 1 and 2, demonstrate that the subset $W=\{f(a,b): a,b \in \Bbb R\}$ is not a subspace of $R(2)[x]$ for the given $f(a,b)$

1) $f(a,b) = (2a-3b+1) + (-2+5b)x + (2a+b)x^2$

I would really appreciate if someone can help me solve this question. My main issue is I don't know what elements to pick for $W$.

Answer 1

Take a loot at the coordinates $$ (2a-3b+1, -2+5b, 2a+b ) = (1, -2, 0)+a(2,0,2)+b(-3,5,1), $$ it is plane in $\mathbb{R}^3$ that does not pass through the origin, i.e., do not contains the $0$ vector. You can check also that it is not closed under scalar multiplication and addition.