can you give me the full explanation process for the problem (a), I can not understand the solution very well.

Give P3(R) the inner product (p, q) = p(x)q(r) dr. (a) Find an orthonormal basis of the subspace U = {p : p(1) = 0} CP3(R). (b) Find an orthonormal basis for U-, and an orthonormal basis for P3 (R) that extends your orthonormal basis for U. (c) Find the polynomial p E P3 (R) such that p(1) = 0 and (1 + 3x - p(x)) dx is as small as possible.Solution. (1) First we take a basis of U. Any polynomial p(a) with p(1) =0 is divisible by a - 1, so of of the form p(a) = (2 - 1)q(2) with degg $ 2. It follows that (on, 82, as} = ((2- 1),2(2 - 1), 2 (2 - 1)) are a basis of U. Next we do Gram-Schmidt to replace this basis of U with an orthonormal basis of U. We use computer algebra to compute the inner products as follows: 1=3-1 V3 e = v3(x - 1) =2(x- 1) - (x(2 - 1), v3(x - 1)) .