range

[man111]

if the range of parameter $t $ in the interval  $\left(0, 2\pi\right)$ satisfying  $\displaystyle \frac{(-2x^2+5x-10)}{(\sin t) x^2 + 2(1+ \sin t )x + 9\sin t +4} > 0 $  for all real values of  $x $ is  $(a,b)$ and  $a+ b=k\pi $ . find $k$

[Руст]

Always $-2x^2+5x-10<0, x^2+2x+9>0.$ Therefore $\sin t (x^2+2x+9)+2x+4$ must be negative. It is always negative, if $\sin t<\frac{-1}{2}$. In this case k is any.

[man111]

$k=3$
thanks pyct