Invariants of the quadrics and Euclidean classification
Let's consider that $4x^2+9y^2+16z^2+12xy+16xz+24yz+2x+4y+6z+1=0$. Classify the quadric.
The matrix associated with the equation of the quadric is: $$\overline{A} = \begin{bmatrix} 4 & 6 & 8 & 1 \\ 6 & 9 & 12 & 2 \\ 8 & 12 & 16 & 3 \\ 1 & 2 & 3 & 1 \end{bmatrix}$$ We are going to calculate, now, its euclidean invariants. $$det(x \cdot I-\overline{A})=x^4-30x^3+15x^2+6x$$ $$det(x \cdot I - A)=x^3-29x^2$$ Therefore, we have : $$\left \{ \begin{array}{l} D_4=0 \\ D_3=-6 \\ D_2=15 \\ D_1=30\end{array} \right.$$ $$\left\{ \begin{array}{l} d_3=0 \\ d_2=0 \\ d_1=29 \end{array} \right.$$
The index of the quadric is $0$ due to the fact that the condition $d_1\cdot d_3 < 0$ and $d_2 < 0$ is not satisfied.
As $D_4=0, d_3=0, d_2=0, D_3=-6$, for the classification algorithm we can see that it is a parabolic cylinder.