1. The problem is to understand and analyze the given equation and the description of the parabola. 2. The equation appears to be a system or matrix-like expression: $$\begin{bmatrix}\{x^2 + 3y + 67\}\end{bmatrix} = d$$ and $$\begin{bmatrix}8 \div 2 + 3\end{bmatrix}$$. 3. First, simplify the scalar expression on the right: $$8 \div 2 + 3 = 4 + 3 = 7$$. 4. So, the equation can be interpreted as $$x^2 + 3y + 67 = d$$ and the scalar value is 7. 5. If we set $$d = 7$$, then the equation becomes $$x^2 + 3y + 67 = 7$$. 6. Rearranging to solve for $$y$$: $$3y = 7 - 67 - x^2$$ $$3y = -60 - x^2$$ 7. Divide both sides by 3: $$y = \frac{-60 - x^2}{3}$$ 8. Using \cancel to show division step: $$y = \cancel{\frac{1}{3}}(-60 - x^2)$$ 9. Simplify: $$y = -20 - \frac{1}{3}x^2$$ 10. This is a parabola opening downwards (since coefficient of $$x^2$$ is negative), but the user described a parabola opening upwards with vertex at approximately (-1.5, -9.5). 11. To match the vertex form, rewrite $$y$$ as: $$y = -\frac{1}{3}x^2 - 20$$ 12. The vertex of $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $$a = -\frac{1}{3}$$ and $$b = 0$$, so vertex $$x = 0$$. 13. The vertex is at (0, -20), which does not match the described vertex (-1.5, -9.5). 14. Therefore, the given equation and the described parabola do not correspond directly. 15. The problem likely involves understanding the parabola with vertex at (-1.5, -9.5) and symmetry about $$x = -1.5$$. 16. The vertex form of a parabola is: $$y = a(x - h)^2 + k$$ where $$(h, k)$$ is the vertex. 17. Using the vertex (-1.5, -9.5), the equation is: $$y = a(x + 1.5)^2 - 9.5$$ 18. Since the parabola opens upwards, $$a > 0$$. 19. Without more points, $$a$$ cannot be determined exactly. 20. Summary: The given matrix-like expression simplifies to a parabola opening downwards with vertex at (0, -20), but the described parabola opens upwards with vertex (-1.5, -9.5). The vertex form for the described parabola is $$y = a(x + 1.5)^2 - 9.5$$ with $$a > 0$$.