1. **State the problem:** Identify the type of conic section represented by the equation $$4X^2 - Y^2 - 8X + 8 = 0$$. 2. **Rewrite the equation:** Group the $X$ terms and isolate constants: $$4X^2 - 8X - Y^2 + 8 = 0$$ 3. **Complete the square for the $X$ terms:** Factor out 4 from the $X$ terms: $$4(X^2 - 2X) - Y^2 + 8 = 0$$ Complete the square inside the parentheses: $$X^2 - 2X = (X - 1)^2 - 1$$ Substitute back: $$4((X - 1)^2 - 1) - Y^2 + 8 = 0$$ 4. **Simplify the equation:** $$4(X - 1)^2 - 4 - Y^2 + 8 = 0$$ $$4(X - 1)^2 - Y^2 + 4 = 0$$ 5. **Isolate terms:** $$4(X - 1)^2 - Y^2 = -4$$ Multiply both sides by $-1$: $$-4(X - 1)^2 + Y^2 = 4$$ Rewrite: $$Y^2 - 4(X - 1)^2 = 4$$ 6. **Divide both sides by 4:** $$\frac{Y^2}{4} - (X - 1)^2 = 1$$ 7. **Identify the conic:** This is the standard form of a hyperbola centered at $(1,0)$ with transverse axis along the $Y$-axis. **Final answer:** The conic is a **hyperbola**.