The sum of all real values of x for which 3 x ^ 2 - 9x + 17 x ^ 2 +...

Given,

$\dfrac{3{x}^{2} - 9x + 17}{{x}^{2} + 3x + 10} = \dfrac{5{x}^{2} - 7x + 19}{3{x}^{2} + 5x + 12}$

$⇒ \dfrac{{x}^{2} + 3x + 10 + 2{x}^{2} - 12x + 7}{{x}^{2} + 3x + 10} = \dfrac{3{x}^{2} + 5x + 12 + 2{x}^{2} - 12x + 7}{3{x}^{2} + 5x + 12}$

$⇒ 1 + \dfrac{2{x}^{2} - 12x + 7}{{x}^{2} + 3x + 10} = 1 + \dfrac{2{x}^{2} - 12x + 7}{3{x}^{2} + 5x + 12}$

$⇒ \left(2{x}^{2} - 12x + 7\right)\left(\dfrac{1}{{x}^{2} + 3x + 10} - \dfrac{1}{3{x}^{2} + 5x + 12}\right) = 0$

$2{x}^{2} - 12x + 7 = 0$ Or $3{x}^{2} + 5x + 12 = {x}^{2} + 3x + 10$

For $2{x}^{2} - 12x + 7 = 0$

Sum of roots$= \dfrac{12}{2} = 6$

Or $3{x}^{2} + 5x + 12 = {x}^{2} + 3x + 10$

$⇒ 2{x}^{2} + 2x + 2 = 0$

$⇒ {x}^{2} + x + 1 = 0$

No real roots as $D = 1 - 4 < 0$