If  $x^2+3x+5=0$  and $a{x}^2+bx+c=0$  have a common root and $a,b,c\in N$  then minimum value of $a+b+c$  is equal to

$x^2+3x+5=0$  has imaginary roots which are complex conjugate.

$\therefore$   $x^2+3x+5=0,a{x}^2+bx+c$  have both roots in common.

$\therefore \frac{a}{1}=\frac{b}{3}=\frac{c}{5}=\lambda$

$\Rightarrow$    $a+b+c=9\lambda ,$          $a,b,c\in N\Rightarrow \lambda \in N$

$\Rightarrow$    $\min .\left(a+b+c\right)=9$