Let B be the basis of P₃ consisting of the Hermite polynomials in Exercise 27, and let p(t) = 7 - 12t - 8t² + 12t³. Find the coordinate vector of p relative to B.

Data from in Exercise 27

The first four Hermite polynomials are 1, 2t, - 2 + 4t², and -12t + 8t³. These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of P₃.