To form a quadratic polynomial $y(x)=ax^2+bx+c$ through the through the three points $(x,y)=(1,4),(2,5),(3,7)$, an expression can be written:
$$y(x)=4\frac{(x-2)(x-3)}{(1-2)(1-3)}+5\frac{(x-1)(x-3)}{(2-1)(2-3)}+7\frac{(x-1)(x-2)}{(3-1)(3-2)}$$
The question is why is the equation written like that? For example, why is the whole expression not expanded? Why not simplify some of the brackets such as $(1−2)$?
Because it makes it very easy to check that the proposed expression indeeds solve the interpolation problem: every term is zero at all but one interpolation point.