1. **State the problem:** Make $x$ the subject of the equation $$y = \frac{5 - 2x}{x + 3}$$. 2. **Formula and rules:** To make $x$ the subject, we need to isolate $x$ on one side of the equation. 3. **Multiply both sides by the denominator to eliminate the fraction:** $$y(x + 3) = 5 - 2x$$ 4. **Expand the left side:** $$yx + 3y = 5 - 2x$$ 5. **Group all $x$ terms on one side and constants on the other:** $$yx + 2x = 5 - 3y$$ 6. **Factor out $x$ on the left side:** $$x(y + 2) = 5 - 3y$$ 7. **Divide both sides by $(y + 2)$ to isolate $x$:** $$x = \frac{5 - 3y}{\cancel{y + 2}} \cancel{\frac{1}{y + 2}}$$ 8. **Final answer:** $$\boxed{x = \frac{5 - 3y}{y + 2}}$$