1. Stating the problem: Solve the equation $$4x^2y + 9y^2x = 36$$ for variables $x$ and $y$. 2. Factor the left side to find a common factor: $$4x^2y + 9y^2x = x y (4x + 9y) = 36$$ 3. Rewrite the equation using the factored form: $$x y (4x + 9y) = 36$$ 4. This equation expresses a relationship among $x$ and $y$. To find explicit solutions, one may express $y$ in terms of $x$ or vice versa. For example, solve for $y$: $$x y (4x + 9y) = 36 \implies y (4x + 9y) = \frac{36}{x}$$ or rearranged: $$4x y + 9 y^2 = \frac{36}{x}$$ 5. This is a quadratic in $y$: $$9 y^2 + 4x y - \frac{36}{x} = 0$$ 6. Use the quadratic formula for $y$, where $a=9$, $b=4x$, and $c=-\frac{36}{x}$: $$y = \frac{-4x \pm \sqrt{(4x)^2 - 4 \times 9 \times (-\frac{36}{x})}}{2 \times 9}$$ 7. Simplify the discriminant: $$(4x)^2 - 4 \times 9 \times (-\frac{36}{x}) = 16 x^2 + 1296/x$$ 8. Final expression for $y$: $$y = \frac{-4x \pm \sqrt{16 x^2 + \frac{1296}{x}}}{18}$$ This formula gives $y$ in terms of $x$, completing the solution set for the equation.