Question

Express P(x)=2-3x+4x² in terms of legend polynomial

Answer 1

Do you mean Legendre polynomials?
P_₀(x) = 1
P_₁(x) = x
P_₂(x) = 1/2 (3x^2-1)
So, x^2 = (2P_₂(x)+1)/3

4x^2-3x+2 = =4*(2P_₂(x)+1)/3 - 3P_₁(x) + 2P_₀(x)