You are looking to find a quiadratic finction in the form

    y = ax^2 + bx + c    (1)

such that its plot goes through three given points.


Notice that two given points, (0,0) and (2,0), have the same (identical) y-values.
Moreover, these y-values are zeros, so these points are two y-intercepts of the parabola.


Hence, the parabola's symmetry axis is half-way between x-coordinates of these points:

     =  = 1.


So, the parabola in the vertex form is  

    y = ,     (2)

where  is the vertex' y-coordinate.


From the other side, you see that the second given point has x-coordinate equal to 1 - hence,
this point is the vertex;  so you conclude that in formula (2)   = -5.


Thus you can write the parabola's expression in the vertex form

    y = .    (3)


Now, to determine the unknown coefficient "a", substitute x= 0 into formula (3) and use
the given info y(0) = 0  (the coordinates of the first point)

    0 = ,

or
    
    0 = a - 5,

which gives you 

    a = 5.


Now the parabola in vertex form is  y = .


To get the standard form, make FOIL

    y(x) = 5x^2 - 10x.


It is your ANSWER.