1. **State the problem:** We are given the equation $$x^2 - x - y^2 - 4y - 5 = 0$$ and want to identify the graph and its key features. 2. **Rewrite the equation by completing the square:** Start with the original equation: $$x^2 - x - y^2 - 4y - 5 = 0$$ Group $x$ and $y$ terms: $$ (x^2 - x) - (y^2 + 4y) = 5 $$ Complete the square for $x$: $$x^2 - x = (x - \frac{1}{2})^2 - \left(\frac{1}{2}\right)^2 = (x - 0.5)^2 - 0.25$$ Complete the square for $y$: $$y^2 + 4y = (y + 2)^2 - 4$$ Substitute back: $$ (x - 0.5)^2 - 0.25 - \left[(y + 2)^2 - 4\right] = 5 $$ Simplify: $$ (x - 0.5)^2 - 0.25 - (y + 2)^2 + 4 = 5 $$ $$ (x - 0.5)^2 - (y + 2)^2 + 3.75 = 5 $$ Subtract 3.75 from both sides: $$ (x - 0.5)^2 - (y + 2)^2 = 5 - 3.75 $$ $$ (x - 0.5)^2 - (y + 2)^2 = 1.25 $$ 3. **Interpret the equation:** This is the standard form of a hyperbola centered at $(0.5, -2)$: $$ (x - h)^2 - (y - k)^2 = c $$ where $c > 0$. 4. **Final answer:** The graph is a hyperbola centered at $(0.5, -2)$ with equation: $$ (x - 0.5)^2 - (y + 2)^2 = 1.25 $$ Note: The user’s earlier step had center at $(1, -2)$ and right side $2$, but completing the square precisely gives center $(0.5, -2)$ and right side $1.25$.