Answer
not a subspace of $\mathbb{P}_{n}$
Work Step by Step
For
$ H=\{p(t)| p(t)=a+t^{2}, a\in R\}$
following the definition of a subspace on p.195,
(a) Is the zero vector in H?
That is, is there an a such that
$a+t^{2}=0$?
By equality of polynomials,
$a+1\cdot t^{2}=0+0t^{2}$,
the answer is NO ($1\neq 0)$.
$H$ is not a subspace of $\mathbb{P}_{n}$
