Find the area of the space which is enclosed by points $(x,y)$ on the plane which also satisfy an equation

Question

Find the area of the space which is enclosed by points $(x,y)$ on the plane which also satisfy the equation $9x^3+9x^2y-45x^2=4y^3+4xy^2-20y^2$

By factorizing we have that $(3x+2y)(3x-2y)(x+y-5)=0$ and hence it is a triangle, which has area of 30. I would never have thought of this on my own. My question is how do you intuitively think of factorizing the equation and suspect that you will receive a triangle? When I read the question, I would never have that that it creates a triangle. Could you please explain to me the intuitive process?

Answer 1

The LHS coefficients are $9,9,-45.$

The RHS coefficients are $4,4, -20.$

The coefficients on both sides are in roughly the same proportions.

LHS = $9(x^3 + x^2 y - 5x^2) = 9x^2(x + y -5).$

RHS = $4(y^3 + x y^2 - 5y^2) = 4y^2(x + y -5).$

LHS - RHS $= (9x^2 - 4y^2)(x + y - 5).$

Both $9x^2$ and $4y^2$ are perfect squares.