Determine if a.H={x,yy=3x−1} is a subspace of R2 b.H={at+bb=8a} is a subspace of P1

Let V be a vector space and $S\subset V$ is said to be a subspace of V if the followings holds
$0\in S$ for u,$v\in S$ and for a scaler k $u+kv\in S$.
(a)Here it is given ${\displaystyle H=\{(x,y):y=3x-1\}.}$ Since (0, 0) is not in H. Thus H is not a subspace of $R^2$.
(b)Here it is given ${\displaystyle H=\{at+b:b=8a\}.}$ Clearly (0, 0) is in H. Let $at+b,ct+d\in H$. Then $b=8aandd=8c$. Now for a scaler k