You didn't really ask a question, but that's a pretty interesting conic.
This is the rare conic that factors on the left:
[tex]2 x^2 + 3 x y + y^2 + 5 x + 2 y - 3 = (x + y + 3) (2 x + y - 1) = 0[/tex]
So this is really two lines,
[tex]x + y + 3 = 0 \textrm{ and } 2x + y - 1 = 0[/tex]
For fun, let's calculate the meet of those two lines:
[tex]y = -x -3[/tex]
[tex]2x + -x -3 - 1 = 0[/tex]
[tex]x = 4[/tex]
[tex]y = -4 -3 = -7[/tex]
This conic is two lines which meet at (4,-7)
The factors of the equation are 4 and -7.
Given
Equation; [tex]\rm 2x^2+3xy+y^2+5x+2y-3=0[/tex]
Factor theorem is used when factoring the polynomials completely.
Therefore,
The factors of the polynomial are;
[tex]\rm 2x^2+3xy+y^2+5x+2y-3=0\\\\2x^2+3xy+5x+y^2+2y-3=0\\\\ 2x^2+xy-x+2xy+y^2-y+6x+3y-3=0\\\\ (x+y+3)(2x+y-1)=0\\\\[/tex]
On solving the equations
[tex]\rm x+y=-3\\\\2x+y=1[/tex]
On subtracting both the equation
[tex]\rm x+y-2x-y=-3-1\\\\-x=-4\\\\x=4[/tex]
Substitute the value of x in the equation first
[tex]\rm x+y=-3\\\\4+y=-3\\\\y=-3-4\\\\y=-7[/tex]
Hence, the factors of the equation are 4 and -7.
To know more about the Factor theorem click the link given below.
https://brainly.com/question/12959513
