Find the area of the space enclosed by points $(x, y)$ such that fullfill the following condition: $9x^3+9x^2y-45x^2=4y^3+4xy^2-20y^2$ Find the area of the space enclosed by points $(x, y)$ such that fullfill the following condition: $9x^3+9x^2y-45x^2=4y^3+4xy^2-20y^2$
My solution goes as follows:
Factorizing we have:
$(9x^2-4y^2)(x+y-5)=0$
$(3x+2y)(3x-2y)(x+y-5)=0$
Hence we get that the vertices of the triangle are points $(-10, 15), (0, 0)$ and $(2, 3)$ and from this we conclude that the area is $30$.
My question is how was I supposed to think of the particular factorization? Can you propose an intuitive method?

Step by step, we have:
$$9(x^3+x^2y-5x^2)=4(y^3+xy^2-5y^2)$$
$$\Rightarrow 9x^2(x + y - 5) = 4y^2 (y + x - 5)$$
and moving everything to one side gives $(9x^2-4y^2)(x + y - 5) = 0$.
