To solve the equation \((2x-1)(x+2) = 2(x+2)\), follow these steps:

1. First, factor out \((x+2)\) from both sides of the equation:

\[
(2x-1)(x+2) = 2(x+2)
\]

2. Divide both sides of the equation by \((x+2)\). Note that we need to consider the special case where \(x+2=0\):

\[
x \neq -2 \implies 2x-1 = 2
\]

3. Solve for \(x\):

\[
2x - 1 = 2
\]

\[
2x = 3
\]

\[
x = \frac{3}{2}
\]

4. Now, we need to consider the special case excluded previously, \((x+2=0)\):

\[
x + 2 = 0
\]

\[
x = -2
\]

Hence, the solution set \(S\) includes both solutions:

\[
S = \left\{ -2, \frac{3}{2} \right\}
\]

So, the correct answer according to the given choices is:

**C. S = {-2, 3/2}**