Using properties of proportion, find the value of ‘x’:

Given,

$\Rightarrow \frac{6x^{2} + 3x - 5}{3x - 5} = \frac{9x^{2} + 2x + 5}{2x + 5}$

Applying componendo and dividendo,

$$\begin{aligned}\Rightarrow \frac{6x^{2} + 3x - 5 + (3x - 5)}{6x^{2} + 3x - 5 -(3x - 5)} = \frac{9x^{2} + 2x + 5 + (2x + 5)}{9x^{2} + 2x + 5 - (2x + 5)} \\[1em] \Rightarrow \frac{6x^{2} + 3x - 5 + 3x - 5}{6x^{2} + 3x - 5 - 3x + 5} = \frac{9x^{2} + 2x + 5 + 2x + 5}{9x^{2} + 2x + 5 - 2x - 5} \\[1em] \Rightarrow \frac{6x^{2} + 6x - 10}{6x^{2}} = \frac{9x^{2} + 4x + 10}{9x^{2}} \\[1em] \Rightarrow \frac{6x^{2} + 6x - 10}{6} = \frac{9x^{2} + 4x + 10}{9} \\[1em] \Rightarrow 9 \times (6x^{2} + 6x - 10) = 6 \times (9x^{2} + 4x + 10) \\[1em] \Rightarrow 54x^{2} + 54x - 90 = 54x^{2} + 24x + 60 \\[1em] \Rightarrow 54x^{2} + 54x - 90 - 54x^{2} - 24x - 60 = 0 \\[1em] \Rightarrow 54x^{2} - 54x^{2} + 54x - 24x - 90 - 60 = 0 \\[1em] \Rightarrow 30x - 150 = 0 \\[1em] \Rightarrow 30x = 150 \\[1em] \Rightarrow x = \frac{150}{30} \\[1em] \Rightarrow x = 5.\end{aligned}$$

Hence, x = 5.