1. **State the problem:** Solve the quadratic equation $4n^2 + 4n + 36 = 0$ by completing the square. 2. **Write the equation:** $$4n^2 + 4n + 36 = 0$$ 3. **Divide the entire equation by 4** to simplify the coefficient of $n^2$ to 1: $$\cancel{4}n^2 + \cancel{4}n + \cancel{4} \times 9 = \cancel{4} \times 0$$ which simplifies to $$n^2 + n + 9 = 0$$ 4. **Isolate the constant term:** $$n^2 + n = -9$$ 5. **Complete the square:** Take half of the coefficient of $n$, which is $\frac{1}{2}$, and square it: $$\left(\frac{1}{2}\right)^2 = \frac{1}{4}$$ Add $\frac{1}{4}$ to both sides: $$n^2 + n + \frac{1}{4} = -9 + \frac{1}{4}$$ 6. **Rewrite the left side as a perfect square:** $$\left(n + \frac{1}{2}\right)^2 = -\frac{36}{4} + \frac{1}{4} = -\frac{35}{4}$$ 7. **Solve for $n$ by taking the square root of both sides:** $$n + \frac{1}{2} = \pm \sqrt{-\frac{35}{4}} = \pm \frac{\sqrt{-35}}{2} = \pm \frac{\sqrt{35}i}{2}$$ 8. **Isolate $n$:** $$n = -\frac{1}{2} \pm \frac{\sqrt{35}i}{2}$$ **Final answer:** $$n = \frac{-1 \pm \sqrt{35}i}{2}$$ This means the solutions are complex numbers because the discriminant is negative.