  The Banach space $\mathcal{P}({}^2X)$ of $2$-homogeneous polynomials on the
Banach space $X$ can be naturally embedded in the Banach space ${{\rm
Lip}_0}(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$.
We investigate whether $\mathcal{P}({}^2X)$ is a complemented subspace of
${{\rm Lip}_0}(B_X)$. This line of research can be considered as a polynomial
counterpart to a classical result by Joram Lindenstrauss, asserting that
$\mathcal{P}({}^1X)=X^*$ is complemented in ${{\rm Lip}_0}(B_X)$ for every
Banach space $X$. Our main result asserts that $\mathcal{P}({}^2X)$ is not
complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$ with
non-trivial type.
