I have two Legendre questions, can anyone help me with either of them?

Suppose

. Let p be a polynomial. Show that

for some polynomial q.

Taking p(x) = 1 this implies

for some polynomial q.

and the other one is...

Use integration by parts to show that

Mr F says: The integration is over the interval (-1, 1).
for all

. Deduce that

is an orthogonal basis of

.

Mr F says: On the interval (-1, 1). may be

or it may be

... Im not sure really which is why im stuck! Both are used in my handout

Mr F says: They are equivalent expressions. The former is the series solution to Legendre's Equation. The latter is Rodrigues' Formula.