Can the substraction of quadratic form ($(x-b)^T A(x-b)$) of two functions be written as another quadratic form plus constant

Question

A function $f: \mathbb R^d \rightarrow \mathbb R$ is said to have quadratic form iff it can be written as $f(x)=(x-b)^T A(x-b)$ for some $A\in \mathbb R^{d\times d}$ and $b \in \mathbb R^d$. If I have two functions having quadratic forms $f(x)=(x-b_1)^T A_1(x-b_1)$ and $g(x)=(x-b_2)^T A_2(x-b_2)$ can I write $h(x):=f(x)-g(x)$ in the form of $(x-b_3)^T A_3(x-b_3) + \alpha_3$? If yes what is the explicit form of $A_3$, $\alpha$, and $b_3$ in terms of $A_1, A_2, b_1, b_2$. If no, why?