How to break the quadratic form $x^TABx + x^TB^TAx$?

Question

Let $Q = AB$, where $A \in \mathbb{R}^{n \times n}$ is symmetric and positive definite, and $B \in \mathbb{R}^{n \times n}$ is square but not necessarily symmetric nor positive definite. Then claim:

$$2x^TQx = 2x^TABx = x^TABx + x^TB^TAx$$

This was used to derived the algebraic Riccati equation. How was this identity derived?

Answer 1

Hint: a scalar equals its transpose and so $(x'ABx)$ equals its transpose. Then it remains to use the symmetry of $A$.