If a, b, c are real, then f(x)= […] is decreasing in

$f'(x)=\begin{vmatrix}1&0&0\\ab&x+b^2&bc\\ac&bc&x+c^2\end{vmatrix}+\begin{vmatrix}x+a^2&ab&ac\\0&1&0\\ac&bc&x+c^2\end{vmatrix}+\begin{vmatrix}x+a^2&ab&ac\\ab&x+b^2&bc\\0&0&1\end{vmatrix}$

$=(x+b^2)(x+c^2)-b^2c^2+(x+a^2)(x+c^2)-a^2c^2+(x+a^2)(x+b^2)-a^2b^2$

$=3x^2+2x(a^2+b^2+c^2)$

$f ( x)$ will be decreasing when $f'(x)<0$

$⇒3x^2+2x(a^2+b^2+c^2)<0⇒x∈(-\frac{2}{3}(a^2+b^2+c^2),0)$