Find the monic polynomial f(x) of minimal degree such that f(1 + i) = f(−1 + i) = 0 and all the coefficients of f(x) are real.

The roots you specfy  will not produce a polynomial  with real coefficients

I think  you must  mean that    f( 1 + i)    = f( 1 - i)  =  0 

So we  have

f(x)  =  (x - ( 1 + i) )   (x - ( 1 - i ) )   =

(x - 1   - i )   (  x - 1  + i)    =

x^2  -x  - ix  - x + 1  + i   + ix - i  - i^2   =

x^2  - 2x  + 2