Sketch the curve with cartesian equation $$ y = \frac{2x(x^2-5)}{x^2-4} $$ and give the equations of the asymptotes and of the tangent to the curve at the origin.

Hence determine the number of real roots of the following equations:

Sketch $C$.

In separate diagrams draw sketches of the curves whose equations are:

1. $ 3x(x^2-5)= (x^2-4)(x+3) $
2. $ 4x(x^2-5)= (x^2-4)(5x-2) $
3. $ 4x^2(x^2-5)^2= (x^2-4)^2(x^2+1) $