Find values of $a$ such that $x^2+ax+a^2+6a \lt 0$ $\forall$ $x \in (1,2)$ Find values of $a$ such that $x^2+ax+a^2+6a \lt 0$ $\forall$ $x \in (1,2)$
My try: Since $y=x^2+ax+a^2+6a$ is an open upward Parabola, the roots $\alpha,\beta$ should be distinct and satisfy $1 \lt \alpha \lt 2$ and $1 \lt \beta \lt 2$ implies both the roots are distinct and lies between $1$ and $2$.
So we have the following conditions:
$1.$ Since it should cut positive Y axis as both the roots are positive we have
 $a^2+6a \gt 0$ 
$2.$ $D \gt 0$ $\implies$ $a^2+8a \lt 0$
$3.$ $f(1) \gt 0$ $\implies$ $a^2+7a+1 \gt 0$
$4.$ $f(2) \gt 0$ $\implies$ $a^2+8a+4 \gt 0$
How to solve all these four inequalities in a simpler way?

Hint: if you reverse the role of $a$ and $x$ you get $$a^2+a(x+6)+x^2<0$$
Solution of coresponding equation (on $a$) is $$a_{1,2} = {-x-6\pm \sqrt{D}\over 2}$$
where $D= -3(x-6)(x+2)>0$ for $x\in(1,2)$. So $$a_1<a<a_2$$ and now you have to find the maximum and minimum of expression(s):
$${-x-6\pm \sqrt{D}\over 2}$$ while $x\in(1,2)$.
