If $$(5x + 2y) : (10x + 3y) = 5 : 9$$, then $$(2x^2 + 3y^2) : (4x^2 + 9y^2) = ?$$

$$(5x + 2y) : (10x + 3y) = 5 : 9$$

$$\frac{5x + 2y}{10x + 3y} = \frac{5}{9}$$

$$\Rightarrow 45x + 18y = 50x + 15y$$

$$\Rightarrow 3y = 5x$$

$$\Rightarrow \frac{y}{x} = \frac{5}{3}$$

$$\Rightarrow \frac{y^2}{x^2} = \frac{25}{9}$$ - - - (1)

now,

$$(2x^2 + 3y^2) : (4x^2 + 9y^2)$$

$$\Rightarrow \frac{2x^2 + 3y^2}{4x^2 + 9y^2} = \frac{x^2(2 + 3\frac{y^2}{x^2})}{x^2(4 + 9\frac{y^2}{x^2})} = \frac{(2 + 3\frac{y^2}{x^2})}{(4 + 9\frac{y^2}{x^2})}$$

From equation(1)-

$$\frac{2 + 3 \times\frac{25}{9}}{4 + 9\times\frac{25}{9}} = \frac{6 + 25}{29 \times 3} = \frac{31}{87}$$

$$(2x^2 + 3y^2) : (4x^2 + 9y^2) = 31 : 87