(x + 1) (x - 3)^2 (2x - 1) is greater than or equal to 0
you have three critical values of x ...
x = -1, x = 3, and x = 1/2
these three values are where the left side of the inequality equals zero.
that means everywhere else, the left side is either positive or negative ... you are looking for those values of x that make the left side positive or zero.
the three critical values break up the values of x into four regions ...
x < -1
-1 < x < 1/2
1/2 < x < 3
x > 3
pick any x-value in each region and "test" it in the original inequality. those regions that make the inequality true are part of your solution set.