Answer
See below
Work Step by Step
Let $u,v \in S$
with $u(x)=a+bx^3+cx^4: a,b,c \in R\\
v(x)=d+ex^3+fx^4: d,e,f \in R$
$\Rightarrow u(x)+v(x)=a+bx^3+cx^4+d+ex^3+fx^4=(a+d)+(b+e)x^3+(c+f)x^4$
$\rightarrow (a+d), (b+e),(c+f) \in R$
Hence $ u(x)+v(x) \in S$.
$\Rightarrow S$ is closed under addition.
Set of Rational numbers is closed under usual addition.
Because our scalars can come from set of real numbers
In particular, let $k$ be a scalar.
and $u(x)=a+bx^3+cx^4: a,b,c \in R$
Obtain: $k.u(x)=k(a+bx^3+cx^4)=ka+kbx^3+kcx^4$
Since $ka,kb,kc \in R$, hence $ku \in S$
Therefore $S$ is closed under scalar multiplication.