Symmetric Matrices & Quadratic Forms · Superchallenge problem

On the 4-dimensional space of real polynomials of degree at most $3$ define the quadratic form $Q(p) = \displaystyle\int_{-1}^{1}p(t)\,p(-t)\,dt$. How many negative eigenvalues does its matrix in the basis $1, t, t^2, t^3$ have?