We have 5 points (x[n],y[n]=f(x[n])) where n goes from 1 to 5.
If the Lagrange polynomial is P(x) then:
P(x)=((x-x2)(x-x3)(x-x4)(x-x5)/((x1-x2)(x1-x3)(x1-x4)(x1-x5)))y1+...
P(x)=10(x-3)(x-5)(x-8)(x-12)/((2-3)(2-5)(2-8)(2-12))+
15(x-2)(x-5)(x-8)(x-12)/((3-2)(3-5)(3-8)(3-12))+
25(x-2)(x-3)(x-8)(x-12)/((5-2)(5-3)(5-8)(5-12))+
40(x-2)(x-3)(x-5)(x-12)/((8-2)(8-3)(8-5)(8-12))+
60(x-2)(x-3)(x-5)(x-8)/((12-2)(12-3)(12-5)(12-8)).
P(x)=(x-3)(x-5)(x-8)(x-12)/18-
(x-2)(x-5)(x-8)(x-12)/6+
25(x-2)(x-3)(x-8)(x-12)/126-
(x-2)(x-3)(x-5)(x-12)/9+
(x-2)(x-3)(x-5)(x-8)/42
All that remains is to put x=4 to find an interpolated value for f(4):
f(4)=-16/9-(-32/3)+800/63-16/9+4/21=20, which does in fact lie between 15 and 25.