1. **State the problem:** Solve the quadratic equation $$5x^2 + 10x - 45 = 0$$ by completing the square. 2. **Divide the entire equation by 5** to simplify the coefficient of $$x^2$$ to 1: $$\frac{5x^2}{5} + \frac{10x}{5} - \frac{45}{5} = \frac{0}{5}$$ which simplifies to $$\cancel{5}x^2 + \cancel{2}x - \cancel{9} = 0$$ actually, correctly: $$x^2 + 2x - 9 = 0$$ 3. **Move the constant term to the right side:** $$x^2 + 2x = 9$$ 4. **Complete the square:** Take half of the coefficient of $$x$$, which is $$2$$, half is $$1$$, then square it: $$1^2 = 1$$. Add this to both sides: $$x^2 + 2x + 1 = 9 + 1$$ 5. **Rewrite the left side as a perfect square:** $$(x + 1)^2 = 10$$ 6. **Take the square root of both sides:** $$x + 1 = \pm \sqrt{10}$$ 7. **Solve for $$x$$:** $$x = -1 \pm \sqrt{10}$$ **Final answer:** $$x = -1 + \sqrt{10} \quad \text{or} \quad x = -1 - \sqrt{10}$$