<s>[INST] Why is the graph of $8x^2+8xy+2y^2+2x+y+5=0$ empty? For the equation
$$8x^2+8xy+2y^2+2x+y+5=0$$
Comparing it with standard equation of conic section we get that
$h^2-ab=0$ and
$$\Delta= \left|\begin{array}{ccc} 
a & h & g \\
h & b & f \\ 
g & f & c \end{array}\right| = 0$$
So as per I know it should represent a pair of straight line which are real and coincident but after plotting it on desmos( an app to plot graphs) it showed nothing. Please help me.
 [/INST] (as Oscar Lanzi started...)
\begin{align}
   2(2x+y)^2 + (2x+y) + 5 &= 0 \\
   16(2x+y)^2 + 8(2x+y) &= -40 \\
   [4(2x+y)+1]^2 &= -39 \\
\end{align}
</s>