Classify the following conic: $$2x^2+4xy+y^2+2x+4=0$$.
See development and solution
Development:
The associated matrix is $$$\displaystyle \overline{A}=\begin{bmatrix} 2 & 2 & 1 \\ 2 & 1 & 0 \\ 1 & 0 & 4 \end{bmatrix}$$$ The associated euclidean invariants are: $$$D_3=det \overline{A}=8-1-16=-9$$$ $$$d_2=2-4=-2$$$ $$$d_1=2+1=3$$$ We do not compute $$D_2$$ because the determinant of the associated matrix is other than zero.
From the classification scheme, as $$D_3\neq0$$ and $$d_2 < 0$$, the conic is a hyperbola.
Solution:
The conic is a hyperbola.