There will be $3$ cases

Case-$I$: Power $2$ goes to $a x^{2}$ and $8$ goes to $'1'$

So cuefficient $ = \frac{{10{\rm{!}}{{\rm{a}}^2}}}{{2{\rm{!}}8{\rm{!}}}}$     …..$(1)$

Case-$II$ : Power $4$ goes to $x $  and $ 6$ goes to $1$

So coefficient $=\frac{10 !}{4 ! 6 !}$      ……$(2)$

Case-$III$ : Power $2$ goes $x$ and $1$ goes to $a x^{2}$ and $7$ goes to $1$

So coefficient $=\frac{10 ! a}{2 ! 1 ! 7 !}$        ……$(3)$

So final coefficient $=(1)+(2)+(3)$

$f(1)=45 a^{2}+360 a+210$

So $f(1)=15\left(3 a^{2}+2.4 a+14\right)$

Minimum value at $a=-\frac{24}{2(3)}=-4$

$\left\{\frac{-b}{2 a}\right\}$