1. **Problem:** Solve the equation $x(x + 2) - 2x(x - 3) = 5 - x^2$. 2. **Formula and rules:** Use distributive property and combine like terms. 3. **Step-by-step solution:** 1) Expand both sides: $$x^2 + 2x - 2x^2 + 6x = 5 - x^2$$ 2) Combine like terms on the left: $$x^2 - 2x^2 + 2x + 6x = -x^2 + 5$$ $$-x^2 + 8x = -x^2 + 5$$ 3) Add $x^2$ to both sides: $$\cancel{-x^2} + 8x + \cancel{x^2} = \cancel{-x^2} + 5 + \cancel{x^2}$$ $$8x = 5$$ 4) Divide both sides by 8: $$x = \frac{5}{8}$$ **Final answer:** $x = \frac{5}{8}$