1. **State the problem:** Solve the differential equation $$2xy - 9x^2 + (2y + x^2 + 1) \frac{dy}{dx} = 0.$$\n\n2. **Rewrite the equation:** Rearrange to isolate $$\frac{dy}{dx}$$:\n$$ (2y + x^2 + 1) \frac{dy}{dx} = 9x^2 - 2xy $$\n$$ \frac{dy}{dx} = \frac{9x^2 - 2xy}{2y + x^2 + 1}.$$\n\n3. **Check if the equation is separable or exact:** The equation is not obviously separable. Let's check if it is exact. Define $$M = 2xy - 9x^2$$ and $$N = 2y + x^2 + 1.$$\n\n4. **Check exactness:** Compute partial derivatives:\n$$ \frac{\partial M}{\partial y} = 2x, \quad \frac{\partial N}{\partial x} = 2x.$$\nSince $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x},$$ the equation is exact.\n\n5. **Find potential function $$\Psi(x,y)$$ such that:**\n$$ \frac{\partial \Psi}{\partial x} = M = 2xy - 9x^2, \quad \frac{\partial \Psi}{\partial y} = N = 2y + x^2 + 1.$$\n\n6. **Integrate $$M$$ with respect to $$x$$:**\n$$ \Psi(x,y) = \int (2xy - 9x^2) dx = yx^2 - 3x^3 + h(y), $$ where $$h(y)$$ is an arbitrary function of $$y$$.\n\n7. **Differentiate $$\Psi$$ with respect to $$y$$:**\n$$ \frac{\partial \Psi}{\partial y} = x^2 + h'(y). $$\nSet equal to $$N$$:\n$$ x^2 + h'(y) = 2y + x^2 + 1 \implies h'(y) = 2y + 1.$$\n\n8. **Integrate $$h'(y)$$:**\n$$ h(y) = y^2 + y + C.$$\n\n9. **Write the implicit solution:**\n$$ \Psi(x,y) = yx^2 - 3x^3 + y^2 + y = C,$$ where $$C$$ is a constant.\n\n**Final answer:** The implicit solution to the differential equation is\n$$ yx^2 - 3x^3 + y^2 + y = C.$$