1. Stating the problem: Find the third derivative $\frac{d^3y}{dx^3}$ of the function $$y=\frac{x^2+2x}{x-1}.$$\n\n2. Simplify the function first by performing polynomial division:\n\n$$\frac{x^2+2x}{x-1} = x+3 + \frac{3}{x-1}.$$\n\n3. Write $y$ as $$y = x + 3 + 3(x-1)^{-1}.$$\n\n4. First derivative using the power rule and chain rule:\n\n$$\frac{dy}{dx} = 1 + 0 + 3 \cdot (-1)(x-1)^{-2} = 1 - \frac{3}{(x-1)^2}.$$\n\n5. Second derivative:\n\n$$\frac{d^2y}{dx^2} = 0 - 3 \cdot (-2)(x-1)^{-3} = \frac{6}{(x-1)^3}.$$\n\n6. Third derivative:\n\n$$\frac{d^3y}{dx^3} = 6 \cdot (-3)(x-1)^{-4} = -\frac{18}{(x-1)^4}.$$\n\nAnswer: $$\boxed{\frac{d^3y}{dx^3} = -\frac{18}{(x-1)^4}}.$$