Question

2x^2+3xy+y^2+5x+2y-3=0

Answer 1

You didn't really ask a question, but that's a pretty interesting conic.

This is the rare conic that factors on the left:

$2 x^2 + 3 x y + y^2 + 5 x + 2 y - 3 = (x + y + 3) (2 x + y - 1) = 0$

So this is really two lines,

$x + y + 3 = 0 \textrm{ and } 2x + y - 1 = 0$

For fun, let's calculate the meet of those two lines:

$y = -x -3$

$2x + -x -3 - 1 = 0$

$x = 4$

$y = -4 -3 = -7$

This conic is two lines which meet at (4,-7)