Gaussian integral with matrix exponents

Question

In general, it is known that for any constants $a$, $b$, $c$ with $a>0$, the gaussian integral which is quadratic in $x$ is given as $$ \int_{-\infty}^{\infty}e^{-ax^{2}+bx+c}dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^{2}}{4a}+c} $$ I would like to extend this to matrices, in general, how do I show that this equation also hold for matrices? Alternately, I'm asking what is the following gaussian integral $$ \int_{-\infty}^{\infty}e^{-\textbf{A}x^{2}+\textbf{B}x+\textbf{C}}dx $$ where $\textbf{A}$, $\textbf{B}$, and $\textbf{C}$ are $N\times N$ matrices?