ANSWER
[tex]a = \frac{4}{5} [/tex]
EXPLANATION
The given parabola has equation,
[tex]y = a(x + 5)(x - 1)[/tex]
The y-intercept is
[tex]y = a(0+ 5)(0- 1)[/tex]
[tex]y = - 5a[/tex]
This gives one vertex of the triangle to be,
[tex](0,-5a)[/tex]
The x-intercept is
[tex]a(x + 5)(x - 1) = 0[/tex]
This implies
[tex](x + 5)(x - 1) = 0[/tex]
[tex](x + 5) = 0 \: or \: (x - 1) = 0[/tex]
[tex]x = - 5 \: or \: x = 1[/tex]
The other two vertices of the triangle is,
[tex](-5,0),(1,0)[/tex]
The height of this triangle is
[tex]h = 5a[/tex]
[tex]Area= \frac{1}{2} \times base \times height[/tex]
[tex]12= \frac{1}{2} \times 6 \times 5a[/tex]
[tex]12 = 15a[/tex]
[tex]a = \frac{12}{15} [/tex]
[tex]a = \frac{4}{5} [/tex]
Since
[tex]a > \: 0[/tex]
it means the parabola opens upwards.
