The vertex of this parabola is at (4, -3). when the x-value is 5, the
y-value is -6. what is the coefficient of the squared expression in the parabola's equation?

Parabola equation with a vertex at (h, k) is given by y = a(x - h)^2 + k
For the give parabola, y = a(x - 4)^2 - 3
y = a(x^2 - 8x + 16) - 3
At (5, -6)
-6 = a((5)^2 - 8(5) + 16) - 3 = a(25 - 40 + 16) - 3 = a - 3
a = -6 + 3 = -3

Therefore, the coefficient of the squared expression is -3.