1. **State the problem:** We need to analyze the equation $$y^3 + y^2 - 5y - x^2 = -4$$ and express it in a form suitable for graphing or further study. 2. **Rewrite the equation:** Add 4 to both sides to isolate terms: $$y^3 + y^2 - 5y - x^2 + 4 = 0$$ 3. **Express $x^2$ in terms of $y$:** $$x^2 = y^3 + y^2 - 5y + 4$$ 4. **Interpretation:** For each value of $y$, $x$ can be found by taking the square root: $$x = \pm \sqrt{y^3 + y^2 - 5y + 4}$$ 5. **Domain considerations:** The expression inside the square root must be non-negative: $$y^3 + y^2 - 5y + 4 \geq 0$$ 6. **Summary:** The graph consists of points $(x,y)$ where $x = \pm \sqrt{y^3 + y^2 - 5y + 4}$ and the radicand is non-negative. This describes a curve symmetric about the $y$-axis. **Final form:** $$x = \pm \sqrt{y^3 + y^2 - 5y + 4}$$