Calculus of a Single Variable, Chapter 4, 4.1, Section 4.1, Problem 23

You need to evaluate the indefinite integral, hence, you need to open the brackets such that:
(x + 1)(3x - 2) = 3x^2 - 2x + 3x - 2 = 3x^2 + x - 2
int(x + 1)(3x - 2)dx = int (3x^2 + x - 2)dx
You need to split the integral:
int (3x^2 + x - 2)dx = int 3x^2 dx + int xdx - int 2dx
You need to use the formula int x^n dx = (x^(n+1))/(n+1) + c
int 3x^2 dx= 3x^3/3 + c => int 3x^2 dx= x^3 + c
int xdx = x^2/2 + c
int 2dx = 2x + c
Gathering the results yields:
int (3x^2 + x - 2)dx = x^3 + x^2/2 - 2x + c
Hence, evaluating the indefinite integral, yields int (3x^2 + x - 2)dx = x^3 + x^2/2 - 2x + c.