= Solution
For a nonnegative integer $\ell$, the chain of the same parity as $\ell$ reaches $n=\ell$, where the numerator of the <recurrence relation> vanishes. The next coefficient and every later coefficient in that chain are zero, producing a polynomial of degree $\ell$. Normalizing it at $x=1$ gives the <Legendre polynomial> $P_\ell$.
The opposite-parity chain never reaches $n=\ell$. Its other possible zero, $n=-\ell-1$, is negative and cannot occur, so a nonzero initial coefficient produces a nonterminating <power series>. It is linearly independent of $P_\ell$ and can be scaled to the <Legendre function of the second kind> $Q_\ell$.
The degree statement requires \b[$\ell\geq0$], as the examples in part (c) indicate. For a negative integer $\ell$, the same equation has the nonnegative degree $-\ell-1$, because $\ell(\ell+1)=(-\ell-1)(-\ell)$. A polynomial cannot literally have negative degree.
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