  We provide a short proof of the theorem that every real multivariate
polynomial has a symmetric determinantal representation, which was first proved
in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity
arises from linear matrix inequalities, J. Funct. Anal. 240 (2006), 105-191. We
then provide an example using our approach and extend our results from the real
field $\mathbb{R}$ to an arbitrary field $\mathbb{F}$ different from
characteristic $2$. The new approach we take is only based on elementary
results from the theory of determinants, the theory of Schur complements, and
basic properties of polynomials.
