If a, b, c be real, then $f(x)=\left|\begin{array}{ccc}x+a^2 & a b & a c \\ a b & x+b^2 & b c \\ a c & b c & x+c^2\end{array}\right|$, is decreasing on 

$\left(-\frac{2}{3}\left(a^2+b^2+c^2\right), 0\right)$

We have,

$f(x)=\left|\begin{array}{ccc} x+a^2 & a b & a c \\ a b & x+b^2 & b c \\ a c & b c & x+c^2 \end{array}\right|$

$\Rightarrow f'(x)=\left|\begin{array}{ccc}1 & 0 & 0 \\ a b & x+b^2 & b c \\ a c & b c & x+c^2\end{array}\right|+\left|\begin{array}{ccc}x+a^2 & a b & a c \\ 0 & 1 & 0 \\ a c & b c & x+c^2\end{array}\right| + \left|\begin{array}{ccc}x+a^2 & a b & a c \\ a b & x+b^2 & b c \\ 0 & 0 & 1\end{array}\right|$

$\Rightarrow f'(x)=3 x^2+2 x\left(a^2+b^2+c^2\right)$

For f(x) to be decreasing, we must have

f'(x) < 0

$\Rightarrow 3 x^2+2 x\left(a^2+b^2+c^2\right)<0 \Rightarrow x \in\left(-\frac{2}{3}\left(a^2+b^2+c^2\right), 0\right)$