3.19: Normal to $\ln$ curve; gradient of $BC$

The normal to the curve $y=\ln\!\left(x^{2}+3\right)+\dfrac{1}{2}x^{2}$ at the point $A$ where $x=0$ meets the $x$-axis at $B$. Point $C$ has coordinates $(0, 3\ln 2)$. Find the gradient of the line $BC$ in terms of $\ln 2$.