1. **Problem:** Solve the equation $$x \cdot (4 + 5x) = (x + 2)^2 - 3(x + 1)(x - 1)$$. 2. **Formula and rules:** Expand both sides and simplify. Use distributive property and difference of squares formula: $a^2 - b^2 = (a-b)(a+b)$. 3. **Step-by-step solution:** - Left side: $$x \cdot (4 + 5x) = 4x + 5x^2$$ - Right side: Expand $(x+2)^2 = x^2 + 4x + 4$ - Expand $3(x+1)(x-1) = 3(x^2 - 1) = 3x^2 - 3$ - So right side: $$x^2 + 4x + 4 - (3x^2 - 3) = x^2 + 4x + 4 - 3x^2 + 3 = -2x^2 + 4x + 7$$ 4. **Set equation:** $$4x + 5x^2 = -2x^2 + 4x + 7$$ 5. **Bring all terms to one side:** $$4x + 5x^2 + 2x^2 - 4x - 7 = 0$$ 6. **Simplify:** $$\cancel{4x} + 5x^2 + 2x^2 - \cancel{4x} - 7 = 0 \implies 7x^2 - 7 = 0$$ 7. **Divide both sides by 7:** $$\frac{7x^2}{\cancel{7}} - \frac{7}{\cancel{7}} = 0 \implies x^2 - 1 = 0$$ 8. **Solve quadratic:** $$x^2 = 1 \implies x = \pm 1$$ **Final answer:** $$x = 1 \text{ or } x = -1$$