The sum of all real values of x for which 3 x 2 - 9 x + 17 x 2 + 3 x + 10 = 5 x 2 - 7 x + 19 3 x 2 + 5 x +…

Given,

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

$\Rightarrow \frac{x^2+3x+10+2x^2-12x+7}{x^2+3x+10}=\frac{3x^2+5x+12+2x^2-12x+7}{3x^2+5x+12}$

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

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

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

For $2x^2-12x+7=0$

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

Or $3x^2+5x+12=x^2+3x+10$

$\Rightarrow 2x^2+2x+2=0$

$\Rightarrow x^2+x+1=0$

No real roots as $D=1-4<0$