To prove that the set of all polynomials with coefficient of $x^2$ equal to $0$ is dense in $C([0,1])$.

Let $C([0, 1])$ denote the metric space of continuous real valued functions on $[0, 1]$ under the supremum metric - i.e., the distance between $f$ and $g$ in $C([0, 1])$ equals $\sup\{|f(x) − g(x)| : x \in [0, 1]\}.$ Let $Q \subseteq C([0, 1])$ be the set of all polynomials in $\mathbb{R}[x]$ in which the coefficient of $x^2$ is $0$. Then $Q$ is dense in $C([0, 1])$.

The set of all polynomials is dense in $C([0,1])$ using Stone-Weierstrass theorem.

If I prove that the set of all polynomials with coefficient of $x^2$ equal to $0$ is dense in the set of all polynomials then I'll be done. How to prove this?

Let $\epsilon >0$. $g(x)=f(x^{1/3})$ defines a continuous function, so there is a polynomial $p$ such that $|f(x^{1/3})-p(x)|<\epsilon$ for all $x$. Hence, $|f(x)-p(x^{3})| <\epsilon$ for all $x$.