1. The problem asks for the coordinates of the vertex of the parabola given by the relation $$y = 3(x - 5)(x + 7)$$. 2. The formula for the vertex of a parabola in factored form $$y = a(x - r_1)(x - r_2)$$ is at the midpoint of the roots $$r_1$$ and $$r_2$$, so the x-coordinate of the vertex is $$x = \frac{r_1 + r_2}{2}$$. 3. Here, the roots are $$r_1 = 5$$ and $$r_2 = -7$$ (note the signs inside the factors). 4. Calculate the x-coordinate of the vertex: $$x = \frac{5 + (-7)}{2} = \frac{5 - 7}{2} = \frac{-2}{2} = -1$$ 5. Substitute $$x = -1$$ back into the original equation to find the y-coordinate: $$y = 3(-1 - 5)(-1 + 7) = 3(-6)(6) = 3 \times (-36) = -108$$ 6. Therefore, the vertex coordinates are $$(-1, -108)$$. 7. Among the options given, the correct vertex is $$(-1, -108)$$.