Answer
not a subspace of $\mathbb{P}_{n}$
Work Step by Step
For
$ H=\{p(t)| p(t)=a+bt+ct^{2}, a,b,c\in \mathbb{Z}\}$
following the definition of a subspace on p.195,
(a) if a=b=c=0$\in \mathbb{Z}$, the zero vector is in H.
(b)
($a_{1}+b_{1}t+c{}_{1}t^{2})+(a_{2}+b_{2}t+c_{2}t^{2})=$
$=(a_{1}+a_{2})+(b_{1}+b_{2})t+(c_{1}+c_{2})t^{2}$
belongs to H, because the set of integers is closed under addition.
(c)
Let $a+bt+ct^{2}, a,b,c\in \mathbb{Z}$,
and, at least one of a,b,c is not zero.
Also, let $d\in \mathbb{R}, d\not\in \mathbb{Z}$.
Then
$da+dbt+dct^{2}\not\in H$,
because at least one coefficient is not an integer.
(H is not closed under multiplication by scalars)
So,
$H$ is not a subspace of $\mathbb{P}_{n}$