f(x) = int (x+3)^2 -3x dx if f(-1) = 0.
f(-1) is given to find the constant of integration and we shall use it after the integration.
int (x+3)^2 - 3x dx
To simplify things we can expand (x+3)^2
(x+3)^2 = x^2+6x+9
int (x+3)^2 - 3x dx
=int x^2 + 6x + 9 - 3x dx
=int x^2 +3x + 9 dx
= x^(2+1)/(2+1) + 3x^(1+1)/(1+1) + 9x + C
=x^3/3 - 3x^2/2 + 9x + C
This is f(x) = x^3/3 + 3x^2/2 + 9x + C
We are given f(-1) = 6 Let us use this
f(-1) = (-1)^3/3 +3(-1)^2/2 + 9(-1) + C
6 = -1/3+3/2 - 9 + C
6 = -2/6+9/6-54/6 + C
6=-47/6+C
6+47/6=C
36/6 + 47/6 = C
83/6 = C
f(x) = x^3/3 +3/2x^2 +9x + 83/6