The equation of a curve is \(x^3 + y^3 + 2xy + 8 = 0\).

(a) Express \(\frac{dy}{dx}\) in terms of \(x\) and \(y\).

The tangent to the curve at the point where \(x = 0\) and the tangent at the point where \(y = 0\) intersect at the acute angle \(\alpha\).

(b) Find the exact value of \(\tan \alpha\).