(i) The set of all vectors in $\mathbb{R}^2$ of the form $(a,b)$ where $a+2b=1$.
(ii) The set of all $2\times2$ matrices whose trace is equal to $0$.
(iii) The set of all polynomials in $P_2$ of the form $a_0+a_1x+a_2x^2$ where the product $a_0a_1a_2\le0$.
(i) Consider $c\cdot{x}$ where $x \in A = \{(a,b) | a+2b = 1, a,b \in \mathbb{R} \}$ and $c \in \mathbb{R}\setminus\{ 1 \}$. Now, $c\cdot{x} = (ca, cb)$ but $ca+ 2(cb) = c$. Hence, $c\cdot{x} \notin A$. Hence, its not closed under scalar multiplication.
(ii) Closed under scalar multiplication. Try to justify yourself.
(iii) Not closed. Consider $p(x) = x^2-x+1$ and $c=-1$.