1. **Problem statement:** 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 terms: $$x^2 + 2x - 2x^2 + 6x = 5 - x^2$$ 2) Combine like terms on the left: $$x^2 - 2x^2 + 2x + 6x = -x^2 + 5$$ 3) Simplify: $$-x^2 + 8x = -x^2 + 5$$ 4) Add $x^2$ to both sides: $$-x^2 + 8x + x^2 = -x^2 + 5 + x^2$$ 5) Simplify: $$8x = 5$$ 6) Divide both sides by 8: $$x = \frac{5}{8}$$ **Final answer:** $x = \frac{5}{8}$