Exercise about finding an orthonormal basis

Question

Consider the vector space $P=\mathbb{R}[x]_{\le3}$ with $*:P\times P \to \mathbb{R}$ defined by $f*g=\int_{-1}^{1}f(t)g(t)dt$.

1. Prove that * is a symmetric bilinear form positive definite.

2. Find an orthonormal basis of (P,*).

I have no problem with the first part, but I'd like to verify if what I did in point 2 is correct.

Consider $B=\{1,x,x^2,x^3\}$, the canonical basis of P, using Gram-Schmidt we find the orthogonal basis $\{1,x, x^2-1/3, x(x^2-3/5)\}$. Hence the orthonormal basis $B'=\{1/\sqrt(2),x\sqrt{3/2}, (x^2-1/3)\sqrt{45/8}, x(x^2-3/5)\sqrt{175/8}\}$.

Is it correct? Thanks in advance