### Instruction: In an equation of the form $k = ax^2 + bx + c$ with $a > 0$, the least possible value of $k$ occurs at $x = -b/(2a)$. In the equation $k = (6x + 12)(x - 8)$, what is the least possible value for $k$?### Response: I notice that the equation is already in factored form, so I can use the zero product property to find the roots of the equation, where $k = 0$.
The roots are $x = -2$ and $x = 8$, which are the x-intercepts of the parabola that represents the equation.
Since the leading coefficient of the quadratic term is positive, the parabola opens upward, and the least possible value of $k$ occurs at the vertex of the parabola, which is halfway between the roots.
The x-coordinate of the vertex is the average of the roots, which is $(-2 + 8)/2 = 3$.
To find the y-coordinate of the vertex, which is the least possible value of $k$, I plug in $x = 3$ into the equation.
I get $k = (6(3) + 12)(3 - 8) = (30)(-5) = -150$.
Therefore, the least possible value of $k$ is -150.