1. We are given the quadratic equation $3x^2 + 48x + 117 = 0$ and asked to solve it by completing the square. 2. First, divide the entire equation by 3 to simplify the coefficients: $$x^2 + 16x + 39 = 0$$ 3. Move the constant term to the right side: $$x^2 + 16x = -39$$ 4. To complete the square, take half of the coefficient of $x$, which is $16$, divide by 2 to get $8$, then square it: $$8^2 = 64$$ 5. Add 64 to both sides to keep the equation balanced: $$x^2 + 16x + 64 = -39 + 64$$ 6. The left side is now a perfect square trinomial: $$(x + 8)^2 = 25$$ 7. Take the square root of both sides: $$x + 8 = \pm 5$$ 8. Solve for $x$: - When $x + 8 = 5$, then $x = 5 - 8 = -3$ - When $x + 8 = -5$, then $x = -5 - 8 = -13$ 9. Therefore, the solutions to the equation are: $$x = -3 \text{ or } x = -13$$