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

Answer 1

Do you mean Legendre polynomials?
P₀(x) = 1
P₁(x) = x
P₂(x) = 1/2 (3x^2-1)
So, x^2 = (2P₂(x)+1)/3

4x^2-3x+2 = =4*(2P₂(x)+1)/3 - 3P₁(x) + 2P₀(x)