f(x)=3(x-4) ​2 +6

Since we have the point (0, -1)   this is the y intercept ....so  c = -1

And we have this system of equations

a(-1)^2 + b(-1) - 1  =  -5

a(2)^2 + b(2) - 1 = 1

Simplifying these, we have

a - b = - 4         (1)

4a + 2b = 2   ⇒  2a + b  = 1       (2)

Add (1) and (2)   and we have that

3a = -3

a = - 1

So  a - b = -4       

-1 - b = - 4

b = 3

So.....the equation of the parabola is

y = -x^2 + 3x - 1

Here's the graph : https://www.desmos.com/calculator/pgv8twlmd1