To solve the equation \(3x + 2 = \frac{2}{x - 1}\), follow these steps:

1. Multiply both sides by \(x - 1\) to eliminate the fraction:

\((3x + 2)(x - 1) = 2\)

2. Expand the left side:

\(3x^2 - 3x + 2x - 2 = 2\)

\(3x^2 - x - 2 = 2\)

3. Bring all terms to one side to form a quadratic equation:

\(3x^2 - x - 4 = 0\)

4. Factor the quadratic equation:

\((3x - 4)(x + 1) = 0\)

5. Solve for \(x\) by setting each factor to zero:

\(3x - 4 = 0\) or \(x + 1 = 0\)

\(3x = 4\) gives \(x = \frac{4}{3}\)

\(x = -1\)

Thus, the solutions are \(x = -1\) and \(x = \frac{4}{3}\).