Answer
$S$ is linearly independent set of vectors.
Work Step by Step
Consider the combination
$$a(x^2+3x+1)+b(2x^2+x-1)+c(4x)=0, \quad a,b,c\in R.$$
Which yields the following system of equations
\begin{align*}
a+2b&=0\\
3a+b+4c&=0\\
a-b&=0.
\end{align*}
The determinant of the coefficient matrix is given by
$$\left| \begin {array}{cccc} 1&2&0\\ 3&1&4\\1&-1&0\end {array}
\right|=12 $$
Since determinant is non zero, hence there exist a unique solution for the above system; that is, the trivial solution,
$$a=0,\quad b=0, \quad c=0.$$
Then, $S$ is linearly independent set of vectors.