Describe the graph for the following equation $(x^2)(x+y+1)=(y^2)(x+y+1)$.

Question

Describe the graph for the following equation $(x^2)(x+y+1)=(y^2)(x+y+1)$.

This is quite straight forward. There is simply an equation with two variables and standard notation is assumed.

Answer 1

$ab = cb \implies (a-c)b = 0 \implies [(a-c = 0) \vee (b = 0)].$

Similarly, $(x^2 - y^2)(x + y + 1) = 0 \implies [(x^2 - y^2 = 0) \vee (x + y + 1 = 0)].$

$x^2 = y^2 \implies [(x = y) \vee (x = -y)].$

So, you have $(3)$ lines.

Answer 2

If $x+y+1\ne0$ we divide both sides by it getting (green lines)

$$x^2-y^2=0$$

which is a conic section, a pair of straight lines.

If $x+y+1=0$ the graph can include it (red line) in the total picture of three factors and three lines representing them:

$$ x+y+1=0,\; x=y,\; x=-y.\;$$