$f(x) = \frac{1}2 x^T Gx + b^T x + c$
Suppose G is not a symmetric matrix. Show that the first term can be rewritten to bring it into a symmetric form. Find the new expression in terms of G.
My approach : Since G is not symmetric we can always write G as a sum of a symmetric (S) and anti-symmetric (A) matrix. Also $x^T A x =0$ . Hence,$ x^T Gx = x^T ( S +A) x = x^T Sx +x^T Ax= x^T Sx $. But what exactly is meant by findng new expression in terms of G is not clear
When $G$ is not symmetric, the coefficient of $x_i x_j$ is $G_{ij} + G_{ji}$. So, you can just replace $G$ by a symmetric matrix $S$ that satisfies $S_{ij}+S_{ji} = G_{ij}+G_{ji}$, i.e. $S_{ij} = \frac 12 (S_{ij} + S_{ji})$. As mentioned in the comments, this means that $S = \frac 12 (G + G^T)$.