# Find the roots of polynomial equation that is from a quadratic matrix

Assume the polynomial equation from a quadratic matrix is written as $$P=XHX^T$$ where $X=[1,x,x^2,\cdots,x^M ]$, $H$ is a symmetric matrix of order $(M+1)$. How do we find the roots of the above polynomial?

If not, when the matrix $H$ is more special, can the question be solved?

An arbitrary polynomial of degree $2M$ can be written in this way: the coefficient of $x^k$ on the right side is the sum of the matrix elements $h_{ij}$ with $i+j = k-2$. There's no special method to find the roots.