One way to answer this is to expand the function:
f(x) = (2x+6)(x-7)
= 2x^2 - 14x + 6x - 42
= 2x^2 - 8x - 42
And then use the formula
to find the x-value of the vertex and then use that to find the y-value of the vertex, while is the minimum value.
x = -(-8) / 2(2)
= 8/4
= 2
f(2) = 2(2)^2 - 8(2) - 42
= 2(4) - 16 - 42
= 8 - 16 - 42
= - 50
So the minimum value is -50, since (2,-50) is the lowest point on the parabola.