Quadratic approximation of vector valued functions

Question

What is the best second order approximation using Taylor's theorem for vector value functions? Specifically in the case when $f: \mathbb{R}^d \to \mathbb{R}$ and then $g= \nabla f$ then the first order approximation by Taylor's theorem is $ g(x) = g(a)+(x-a) \cdot \nabla g(a) + ... = g(a) + (x-a) \cdot H(f)(a)+... $ what would be the third term in this approximation?