Consider \(W = \left \{ a + bt + ct^{2}; a + b + c = 0 \right \}\) Yes.W is a subspace of \(\displaystyle{P}_{{{2}}}.\) Since \(\displaystyle{0}\epsilon{W}.\) If \(\displaystyle{x}+{y}{t}+{z}{t}^{{{2}}}{\quad\text{and}\quad}{l}+{m}{t}+{n}{t}^{{{2}}}\) are elements of \(\displaystyle{P}_{{{2}}},\) then \(\displaystyle{\left({x}+{l}\right)}+{\left({y}+{m}\right)}{t}+{\left({z}+{n}\right)}{t}^{{2}}\epsilon{W}.\) So w is closed under vector addtion. Let \(\displaystyle{r}\epsilon{\mathbb{{{R}}}}\) and \(\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\epsilon{W}\) then \(\displaystyle{r}{\left({a}+{b}{t}+{c}{t}{2}\right)}={r}{a}+{r}{b}{t}+{r}{c}{t}^{{2}}\epsilon{W}.\) Therefore w is closed under scalar multiplication . Therefore W is a subspace of \(\displaystyle{P}_{{2}}.\)