= Solution
Part (a) makes $\lambda_k<0$ for all sufficiently large $k$, while part (b) would make $\lambda_k\geq0$ whenever $P^{(k)}$ were nonzero. Hence some derivative of $P$ vanishes identically, so $P$ is a <polynomial>.
Let its degree be $n$. Then $P^{(n)}$ is a nonzero constant. Substituting $k=n$ into the differentiated equation annihilates both derivative terms and gives
$$
\lambda_nP^{(n)}=0.
$$
Thus $\lambda_n=0$, and
$$
\boxed{\lambda=n(n+1)},\qquad n=0,1,2,\ldots.
$$
The nonzero regular solutions are therefore constant multiples of the <Legendre polynomial> of degree $n$.
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