Answer
True
Work Step by Step
Consider the vector space $P(R)$
Let $S=\{p_1(x),p_2(x),...p_n(x)\}$ be a infinite set spans $P$
Then let $M=max\{degree(p_i(x))\}$
If $P= span (S)$, there will be no polynomial on $P$ whose degree is greater than $M$. Thus, $p(x)=x^{M+1}$ is not in $P$.
Hence, the vector space $P(R)$ of all polynomials with real coefficients cannot be spanned by a finite set $S$
