  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 ${\rm
Lip}_0(B_X)$. Especially important for us is when $X$ is the separable Hilbert
space, in which case the question is related to an open question posed by
Gilles Godefroy asking whether ${\rm Lip}_0(\ell_2)$ has the approximation
property. Moreover, our results can be interpreted as failure of a polynomial
counterpart to a classical result by Joram Lindenstrauss, asserting that
$\mathcal{P}({}^1X)=X^*$ is complemented in ${\rm Lip}_0(B_X)$.
