Recall partial fraction decomposition $($this is a technique used to integrate rational functions and used in Laplace Transformations$).$ Given the partial fraction decomposition

$\frac{3x^3+4x^2-6x}{(x^2+2x+2)(x^2-1)}=\frac{Ax+B}{x^2+2x+2}+\frac{C}{x-1}+\frac{D}{x+1}$

find the values of the constants $A,B,C,$ and $D$ in the following method: Multiply through by the least common denominator, equate the coefficients of powers of $x$ to set up a linear system of equations, set up the corresponding augmented matrix, and use technology to solve the augmented matrix, and find the constants $A,B,C,$ and $D.$