(a)[3]
Determine the quotient and remainder when $x^3 + 5x^2 - 2x - 15$ is divided by $x^2 - 3$.
(b)[5]
The variables $x$ and $y$ satisfy the differential equation $\frac{dy}{dx} = \frac{x^3 + 5x^2 - 2x - 15}{6y(x^2 - 3)}$. You are also told that $y = 2$ when $x = 2$. Solve the differential equation to find an expression for $y^2$ in terms of $x$.