1. **State the problem:** Make $x$ the subject of the equation $$y = \sqrt[3]{\frac{6 + 5x}{x + 4}}.$$ 2. **Rewrite the equation:** Cube both sides to eliminate the cube root: $$y^3 = \frac{6 + 5x}{x + 4}.$$ 3. **Cross-multiply:** Multiply both sides by $(x + 4)$ to get rid of the denominator: $$y^3 (x + 4) = 6 + 5x.$$ 4. **Expand the left side:** $$y^3 x + 4 y^3 = 6 + 5x.$$ 5. **Group terms with $x$ on one side:** $$y^3 x - 5x = 6 - 4 y^3.$$ 6. **Factor out $x$ on the left:** $$x (y^3 - 5) = 6 - 4 y^3.$$ 7. **Solve for $x$:** $$x = \frac{6 - 4 y^3}{y^3 - 5}.$$ **Final answer:** $$\boxed{x = \frac{6 - 4 y^3}{y^3 - 5}}.$$