We are asked to expand the function ( f(x) = x^4 - 1 ) in a series of the form ( \sum_{k=0}^{\infty} A_k P_k(x) ), where ( P_k(x) ) are Legendre polynomials. The standard approach is to use the orthogonality of Legendre polynomials over ([-1,1]) with weight 1: [ \int_{-1}^1 P_n(x) P_m(x) , dx = \frac{2}{2n+1} \delta_{nm}. ] The coefficients ( A_k ) are given by: [ A_k = \frac{2k+1}{2} \int_{-1}^1 f(x) P_k(x) , dx. ] We compute the first few coefficients. The Legendre polynomials are: [ P_0(x) = 1, \quad P_1(x) = x, \quad P_2(x) = \frac{3x^2-1}{2}, \quad P_3(x) = \frac{5x^3-3x}{2}, \quad P_4(x) = \frac{35x^4 - 30x^2 + 3}{8}. ] --- Step 1: Compute (A_0): [ A_0 = \frac{1}{2} \int_{-1}^1 (x^4 - 1) , dx = \frac{1}{2} \left[ \int_{-1}^1 x^4 dx - \int_{-1}^1 1 dx \right]. ] [ \int_{-1}^1 x^4 dx = 2 \int_0^1 x^4 dx = 2 \cdot \frac{1}{5} = \frac{2}{5}, \quad \int_{-1}^1 1 dx = 2. ] [ A_0 = \frac{1}{2} \left( \frac{2}{5}