Answer
Slope = $0$
Work Step by Step
We use implicit differentiation:
$y^3 + yx^2 + x^2 -3y^2 = 0$
$3y^2 \frac{dy}{dx} + x^2 * \frac{dy}{dx} + 2xy + 2x - 6y * \frac{dy}{dx} = 0$
$(3y^2 -6y+x^2) \frac{dy}{dx} = -2x-2xy$
$ \frac{dy}{dx} = \frac{-2x-2xy}{3y^2-6y+x^2}$
$ \frac{dy}{dx}|_{(0,3)} = 0$