Identify the conic section $$$\left(x - 5\right) \left(x + y\right) + 3 \left(x + y\right)^{2} = 0$$$

Identify and find the properties of the conic section $$$\left(x - 5\right) \left(x + y\right) + 3 \left(x + y\right)^{2} = 0$$$.

The general equation of a conic section is $$$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$$$.

In our case, $$$A = 4$$$, $$$B = 7$$$, $$$C = 3$$$, $$$D = -5$$$, $$$E = -5$$$, $$$F = 0$$$.

The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = 0$$$.

Next, $$$B^{2} - 4 A C = 1$$$.

Since $$$\Delta = 0$$$, this is the degenerated conic section.

Since $$$B^{2} - 4 A C \gt 0$$$, the equation represents two distinct intersecting lines.

$$$\left(x - 5\right) \left(x + y\right) + 3 \left(x + y\right)^{2} = 0$$$A represents a pair of the lines $$$y = \frac{5}{3} - \frac{4 x}{3}$$$, $$$y = - x$$$A.

General form: $$$4 x^{2} + 7 x y - 5 x + 3 y^{2} - 5 y = 0$$$A.

Factored form: $$$\left(x + y\right) \left(4 x + 3 y - 5\right) = 0$$$A.

Graph: see the graphing calculator.