what is the polynomial f(x)=x^3-x^2+ax+b if f(0)=4 and x=2 is the root of f?

We'll recall the reminder theorem that states the
followings;

f(a) = 0 <=> a is the root of the
polynomial

f(a) = r, where r is the reminder of
polynomial.

According to all the above, we'll
have:

f(2) = 0 => 2^3 - 2^2 + 2a + b =
0

8 - 4 + 2a + b = 0

2a + b =
-4 (1)

f(0) = 4 => b = 4
(2)

We'll replace (2) in
(1):

2a + 4 = -4

2a = -8
=> a = -4

The requested polynomial is:
f(x) = x^3 - x^2 - 4x + 4.