I would like to find coefficients to polynomials, for example,

f(x) = ax^2 + bx + c
f(0) = 3
f(1) = 4
f(2) = 7

should solve for a, b, c such that f(x) = x^2 + 3 (a=1, b=0, c=3).

Another example:

f(x)  =  ax^3 + bx^2 + cx + d
f(2)  =  5
f(0)  =  1
f'(2) = 22
f'(0) =  2

The solution is 5x^3 - 10x^2 + 2x + 1.

I can solve these kinds of problems by substituting x in the function or its derivative which will give me four linear equations with a, b, c and d, which in turn can easily be solved.

Is it possible to directly express this kind of problem using the WolframAlpha website, without writing down the intermediate manual steps?