Let $\mathcal{P}_n$ be the vector space of all polynomials of degree $n$ or less in the variable $x$. Let $D^2 : \mathcal{P}_4 \to \mathcal{P}_2$ be the linear transformation that takes a polynomial to its second derivative. That is, $D^2(p(x)) = p''(x)$ for any polynomial $p(x)$ of degree $4$ or less.

A basis for the kernel of $D^2$ is $\big\lbrace$ $\big\rbrace$. Enter a polynomial or a comma separated list of polynomials.

A basis for the image of $D^2$ is $\big\lbrace$ $\big\rbrace$. Enter a polynomial or a comma separated list of polynomials.